From 3dc01c5a75609f26de6cff3e05cad687087665fe Mon Sep 17 00:00:00 2001 From: Ian Jauslin Date: Thu, 6 Feb 2025 15:10:06 -0500 Subject: [PATCH] Update to v1.0: Major overhaul: All sections have been changed and improved. Added: Computation of the condensate fraction Added: Discussion and plots of correlation function and momentum distribution Added: Section on numerical computation Added: Appendices on functional analysis Added: Detailed analysis of the operator K_e Change: Parameters for numerical simulations. Change: Renamed "Simplified approach" to "Lieb's simplified approach" Added: More bibliography Added: Acknowledgements --- Changelog | 22 + Jauslin_2023c.tex | 2606 +++++++++++++---- README | 9 +- bibliography/bibliography.tex | 54 +- figs/condensate.fig/2020-08-27+0.5.dat | 8 - figs/condensate.fig/2020-10-15+energy.dat | 15 + figs/condensate.fig/Makefile | 8 +- figs/condensate.fig/bigeq.dat | 200 +- figs/condensate.fig/condensate.gnuplot | 19 +- figs/condensate.fig/medeq.dat | 200 +- figs/condensate.fig/simpleq.dat | 200 +- .../2020-08-27+correlation.dat | 102 + .../2020-10-01+correlation16-0001.dat | 102 + figs/correlation.fig/Makefile | 43 + figs/correlation.fig/bigeq0001.dat | 100 + figs/correlation.fig/bigeq02.dat | 100 + figs/correlation.fig/correlation0001.gnuplot | 30 + figs/correlation.fig/correlation02.gnuplot | 28 + figs/correlation.fig/medeq0001.dat | 100 + figs/correlation.fig/medeq02.dat | 100 + figs/correlation.fig/simpleq0001.dat | 100 + figs/correlation.fig/simpleq02.dat | 100 + figs/energy.fig/Makefile | 2 +- figs/energy.fig/bigeq.dat | 200 +- figs/energy.fig/medeq.dat | 120 +- figs/energy.fig/simpleq.dat | 120 +- libs/ian.cls | 161 +- libs/problemset.sty | 2 +- 28 files changed, 3681 insertions(+), 1170 deletions(-) delete mode 100644 figs/condensate.fig/2020-08-27+0.5.dat create mode 100644 figs/condensate.fig/2020-10-15+energy.dat create mode 100644 figs/correlation.fig/2020-08-27+correlation.dat create mode 100644 figs/correlation.fig/2020-10-01+correlation16-0001.dat create mode 100644 figs/correlation.fig/Makefile create mode 100644 figs/correlation.fig/bigeq0001.dat create mode 100644 figs/correlation.fig/bigeq02.dat create mode 100644 figs/correlation.fig/correlation0001.gnuplot create mode 100644 figs/correlation.fig/correlation02.gnuplot create mode 100644 figs/correlation.fig/medeq0001.dat create mode 100644 figs/correlation.fig/medeq02.dat create mode 100644 figs/correlation.fig/simpleq0001.dat create mode 100644 figs/correlation.fig/simpleq02.dat diff --git a/Changelog b/Changelog index 091e29a..bbe4c3b 100644 --- a/Changelog +++ b/Changelog @@ -1,3 +1,25 @@ +v1.0: + * Major overhaul: All sections have been changed and improved. + + * Added: Computation of the condensate fraction + + * Added: Discussion and plots of correlation function and momentum distribution + + * Added: Section on numerical computation + + * Added: Appendices on functional analysis + + * Added: Detailed analysis of the operator K_e + + * Change: Parameters for numerical simulations. + + * Change: Renamed "Simplified approach" to "Lieb's simplified approach" + + * Added: More bibliography + + * Added: Acknowledgements + + v0.2: * Added: Discussion of uniqueness of the solution of the Simple Equation diff --git a/Jauslin_2023c.tex b/Jauslin_2023c.tex index c9fd15f..554a960 100644 --- a/Jauslin_2023c.tex +++ b/Jauslin_2023c.tex @@ -27,9 +27,9 @@ \hbox{} \hfil{\bf\LARGE -Bose-Einstein condensation\par +An introduction to\par \vskip10pt -\hfil and the Simplified Approach to interacting Bosons +\hfil Lieb's Simplified approach to the Bose gas } \vskip20pt @@ -42,68 +42,113 @@ Bose-Einstein condensation\par \hfil {\bf Abstract}\par \medskip -These are lecture notes for a short course on the Simplified Approach to interacting Bosons. -We discuss properties of systems of many interacting Bosons, such as gasses of Helium atoms. -We pay special attention to the phenomenon of Bose-Einstein condensation, which is a quantum phase in which a positive fraction of particles are all in the same state. -We first give an overview of Bose-Einstein condensation in non-interacting systems. -Next, we introduce the Simplified Approach, and prove some of its properties. +This is a book about Lieb's Simplified approach to the Bose gas, which is a family of effective single-particle equations to study the ground state of many-body systems of interacting Bosons. +It was introduced by Lieb in 1963, and recently found to have some rather intriguing properties. +One of the equations of the approach, called the Simple equation, has been proved to make a prediction for the ground state energy that is asymptotically accurate both in the low- and the high-density regimes. +Its predictions for the condensate fraction, two-point correlation function, and momentum distribution also agree with those of Bogolyubov theory at low density, despite the fact that it is based on ideas that are very different from those of Bogolyubov theory. +In addition, another equation of the approach called the Big equation has been found to yield numerically accurate results for these observables over the entire range of densities for certain interaction potentials. +\smallskip + +This book is an introduction to Lieb's Simplified approach, and little background knowledge is assumed. +We begin with a discussion of Bose gases and quantum statistical mechanics, and the notion of Bose-Einstein condensation, which is one of the main motivations for the approach. +We then move on to an abridged bibliographical overview on known theorems and conjectures about Bose gases in the thermodynamic limit. +Next, we introduce Lieb's Simplified approach, and its derivation from the many-body problem. +We then give an overview of results, both analytical and numerical, on the predictions of the approach. +We then conclude with a list of open problems. \vskip20pt +This is a preprint of the following work: I. Jauslin, {\it An Introduction to Lieb's Simplified Approach}, 2025, Springer. +It is the version of the author's manuscript prior to acceptance for publication and has not undergone editorial and/or peer review on behalf of the Publisher (where applicable). +The final authenticated version is available online at: + +\hfil{\color{blue}\href{http://dx.doi.org/10.1007/978-3-031-81393-1}{http://dx.doi.org/10.1007/978-3-031-81393-1}} + +\vskip20pt + + \tableofcontents -\vskip40pt +\vfill +\eject \setcounter{page}1 \pagestyle{plain} \section{Introduction} \indent -This course is about systems of many interacting Bosons. -The first question that we will address is: why should we study interacting Bosons in the first place? -Our answer is threefold. +This book is about interacting Bosons. +The notion of a ``Boson'' was introduced a hundred years before this book was written, in 1924\-~\cite{Bo24,Ei24}, and has garnered much attention, from physicists and mathematicians alike. +In the mathematical physics community, interest in systems of interacting Bosons ballooned at the end of the 1990's, and their study has become one of the major research areas of the field. +As such, much has been written about such systems: research and review articles as well as books. +Nevertheless, despite much work and significant advances, one of the core, foundational questions on Bose gases is still lacking a satisfactory answer: there is still no mathematical proof that interacting Bosons form condensates (at least not in realistic models). +This has been a well-known open question for several decades, and many approaches have been attempted to solve it, without success. +%It may seem that we have reached a point where new ideas are needed. +This book is about a new idea to study interacting Bosons: Lieb's Simplified approach. +(To be exact, the idea is old, but the realization of its significance is new.) +Now, the approach has not, as of this writing, helped solve the problem of condensation, and it may never do so. +But it is a novel approach to the study of interacting Bosons, and it has already yielded some intriguing results. +The aim of this book is to introduce Lieb's Simplified approach and the ideas behind it, in a way that is more pedagogical than the research articles currently available in the literature. +\bigskip + +\indent +This book is based on a series of lectures I gave at a summer school organized by Jake Fillman, at Texas State University, in the summer of 2023. +That course was aimed at undergraduate and young graduate students, and thus required very little prior knowledge on Bose gases. +We will take a similar approach here, and start our discussion at the very beginning. +Readers who are already familiar with Bose gases may prefer to skip to Chapter\-~\ref{sec:simplified_def} where Lieb's Simplified approach is introduced. +\bigskip + +\indent +And so, we start from the very beginning with what may be the most fundamental and important question about systems of interacting Bosons: why are they interesting, and why should we spend time and effort studying them? +There are many possible answers to this question; let us discuss three which I have found to be personally motivating. First, Bosons exist, and if we want to understand the natural world, we are going to have to study them. Second, systems of many interacting Bosons exhibit interesting and non-trivial physical behavior. -Third, they are very challenging to study mathematically, which means that we need to develop new mathematical tools. -Let us expand on each of these points. +Third, they are very challenging to study mathematically, which means that new mathematical tools need to be developed. +Let us now expand on each of these points. \bigskip \point -One of the great successes of science is the understanding that matter is made of particles, and the properties of matter can be understood from the microscopic dynamics of the underlying particles. -But can we actually do this? -The branch of physics that addresses this question is statistical mechanics, which aims to compute the behavior of matter starting from the microscopic laws of motion guiding its constituents particles. -These laws of motion are quantum mechanical, and, in quantum mechanics there are two types of particles: Fermions (such as electrons, protons, neutrinos, etc) or Bosons (such as Helium atoms, photons, Higgs particles, etc). -Here, we will focus on Bosons. -A good example to keep in mind throughout this course is a gas of Helium atoms in a container. -So Bosonic systems exist. +One of the great successes of modern (and not-so-modern) science is the understanding that matter is made of particles, and the properties of matter can be understood from the microscopic dynamics of the underlying particles. +This is a powerful idea: the macroscopic world is very varied: on the face of it, there seems to be very little in common between a tree, a car battery, and a tectonic plate, but they are all made of the same stuff: protons, neutrons, electrons, and assorted particles. +If we can understand the behavior of protons, neutrons, and electrons, we should, at least in principle, be able to understand the properties of all matter. +There is one problem though: deriving macroscopic behaviors from microscopic properties is rather difficult to do in practice. +The branch of physics dedicated to understanding how to do so is called ``Statistical Mechanics'', and aims to compute the behavior of observable matter starting from the microscopic laws of motion guiding its constituents particles. +These laws of motion are quantum mechanical, and, in quantum mechanics there are two types of particles: Fermions (such as electrons, protons, neutrinos, etc) and Bosons (such as photons, Higgs particles, Helium-4 atoms, etc). +(At least this is the case in three dimensions: there are particles that are neither Fermions nor Bosons in two dimensions.) +As we will see, things are complicated, and whereas it would be great to study all types of particles and their mixtures, it is difficult enough to consider even the simplest case in which the particles are all identical Bosons. +And so we will focus here exclusively on systems of identical Bosons. +Such systems exist in nature, in fact, Bosons were introduced to study the photons emanating from matter that is heated (such as the filament in a light bulb)\-~\cite{Bo24,Ei24}. +Another example, which is more relevant to the discussion in this book, is a gas of Helium-4 atoms. \bigskip \point -Helium has many interesting properties, but perhaps the most intriguing is that, when the temperature is small enough (below $2\mathrm{K}$, that is, $-271^\circ\mathrm{C}$ or $-456^\circ\mathrm{F}$), it forms a superfluid phase, which looks like a liquid phase, but it can flow without any viscosity, which leads to some fairly surprising behaviors. -For instance, when a superfluid is put into a container, capillary forces attract it to the edges of the container, but, since there is no viscosity to compensate for them, these forces pull the superfluid all the way up the wall and trickling down the sides. -Another interesting aspect of superfluidity is that classical models do not exhibit superfluidity, so this phase is intrinsically quantum. +Helium has many interesting properties. +One of the most intriguing is that, when the temperature is small enough (below $2\mathrm{K}$, that is, $-271^\circ\mathrm{C}$ or $-456^\circ\mathrm{F}$), it forms a superfluid phase, which looks like a liquid phase, but it can flow without any viscosity\-~\cite{Ka38}, which leads to some fairly surprising behaviors. +For instance, when a superfluid is placed inside a container, capillary forces attract it to the edges of the container, but, since there is no viscosity to compensate for them, these forces pull the superfluid all the way up the wall and trickling down the sides. +This has been observed in experiments: superfluid helium is poured into a glass container, and it can be seen to drip out from the bottom of the container\-~\cite{RS39}. +In addition, because of the lack of viscosity, superfluid helium can pass through tiny little holes in materials. \bigskip \indent -Superfluids are very interesting, but not so simple to study. -Instead, we will simplify things and focus on a different, but also inherently quantum phase: the Bose-Einstein condensate. -This phase was first predicted by Bose\-~\cite{Bo24} and Einstein\-~\cite{Ei24} nearly a century ago, and occurs when most of the particles are all in the same quantum state. +Understanding superfluidity from microscopic dynamics is not an easy task (see\-~\cite{LSe05} for an overview of mathematical results). +There are effective theories that are designed to explain the phenomenon (see\-~\cite{PS16} and references therein), but it is still an open problem to prove that these are linked to realistic microscopic models. +One thing that is clear, however, is that the phenomenon of superfluidity can only arise in quantum models: if the microscopic dynamics were classical, superfluidity would be impossible. +In other words, superfluidity is intrinsically a quantum phenomenon. +Another phenomenon that is intrinsically quantum and has been observed in systems of interacting Bosons is Bose-Einstein condensation\-~\cite{AEe95,DMe95}. +A Bose-Einstein condensate is a phase in which a majority of the particles are all in the same state. In other words, most of the particles are doing the same thing at the same time. -As the superfluid phase, this phenomenon does not occur for classical particles, so this is an inherently quantum phase. -In fact, it seems somewhat related to the superfluid: if the particles are doing the same thing, why would there be viscosity? -(The actual picture is much more complicated than that however.) -\bigskip - -\indent -Thus, systems of many Bosons exhibit non-trivial physical behavior. +There is an idea that Bose-Einstein condensation should be related to superfluidity: after all, if the particles are doing the same thing, there should be no friction between particles, and thus no viscosity. +However, the picture is more complicated than that, and the nature of the link between Bose-Einstein condensation and superfluidity is still unclear. +(It is worth noting, for instance, that there are cases in which superfluidity is believed to occur without Bose-Einstein condensation, for instance, in two dimensional models at positive temperature). +But Bose-Einstein condensation is interesting in its own right: it is an inherently quantum phase of matter that can be realized in experiments, in which most particles are in the same exact state. \bigskip \point Bose-Einstein condensation has had a resurgence in popularity after 1995, when it was first observed experimentally\-~\cite{AEe95,DMe95}. -In particular, the mathematical physics community set off to solve an important problem: the works of Bose and Einstein are done in a much simplified model, in which particles are assumed not to interact with each other. +In particular, the mathematical physics community set off to solve an important problem: the works of Bose\-~\cite{Bo24} and Einstein\-~\cite{Ei24} that first predicted the phenomenon use a much simplified model, in which particles are assumed not to interact with each other. This makes things much simpler: if the particles are independent, it suffices to understand how a single particle behaves, since the situation is identical for all the others. However, in the presence of interactions, no one has yet been able to prove that Bose-Einstein condensation even occurs (and real Bosonic systems have interactions, except for photons, but for those, we don't even know how to write down an appropriate wave equation\-~\cite{Op31}). -(Here, the word ``prove'' is key: most everyone accepts Bose-Einstein condensation occurs in interacting systems, but we don't have a mathematical proof yet.) -In fact, before 1998, there was no complete mathematical handle on pretty much any of the properties of interacting Bosonic gasses. +Here, the word ``prove'' is key: most everyone accepts Bose-Einstein condensation occurs in interacting systems, but we don't have a mathematical proof yet. +In fact, before 1998, there was no complete mathematical handle on pretty much any of the properties of interacting Bosonic gases. In 1998, Lieb and Yngvason\-~\cite{LY98} computed the asymptotic behavior at low density of the ground-state energy (see\-~(\ref{lhy_e})) and this started a flurry of activity. Back in 1957, Lee, Huang and Yang\-~\cite{LHY57} had made a series of predictions, one of which was a low-density expansion for the ground-state energy that went one step further that what Lieb and Yngvason proved in 1998. The mathematical physics community rose to the challenge, and embarked upon a twenty year effort to prove the Lee-Huang-Yang formula. @@ -112,50 +157,70 @@ This was finally accomplished in 2020, after a long series of works culminating \indent However, these successes only concern the ground-state energy. -In particular, there still is no proof of Bose-Einstein condensation. +In particular, there still is no proof of Bose-Einstein condensation in (continuum) interacting models. The results on the ground-state energy were obtained by finding a robust mathematical basis for {\it Boglyubov theory}. -Bogolyubov theory\-~\cite{Bo47} is an approach that was introduced by Bogolyubov in the 1940's to understand the low-density properties of Bose gasses. +Bogolyubov theory\-~\cite{Bo47} is an approach that was introduced by Bogolyubov in the 1940's to understand the low-density properties of Bose gases. One can do a lot with Bogolyubov theory, and many predictions can be made from it\-~\cite{LHY57}. -However, to this day, only its approach to the ground-state energy has been made rigorous. +However, to this day, only its approach to the ground-state energy has been used in mathematical proofs (at least for systems in the thermodynamic limit: there are proofs in other settings, as we will briefly discuss in Chapter\-~\ref{sec:bose_results}). \bigskip \indent -In this course, we will present a different approach, called the {\it Simplified Approach}. -The Simplified Approach was first introduced by Lieb in 1963\-~\cite{Li63}, but after a short series of papers by Lieb and coauthors\-~\cite{LS64,LL64}, it has largely been forgotten. -We have recently had a fresh look at this approach, in a collaboration with Carlen and Lieb\-~\cite{CJL20,CJL21,CHe21,Ja22,Ja23,Ja23b} and found that the predictions of the Simplified Approach far exceeded our expectations. -Indeed, we found that it is as good as Bogolyubov theory at low densities\-~\cite{CJL20,CJL21} (at least as far as the ground state is concerned), it also reproduces the behavior of the Bose gas at high densities\-~\cite{CJL20}. -We have even found very good agreement for some systems {\it over the entire range of densities}\-~\cite{CHe21,Ja23b}. -\bigskip - -\indent -The Simplified Approach consists in replacing the original problem that involves an infinite number of particles with a non-linear, non-local equation for a single particle. +In this book, we will present a different approach, called {\it Lieb's Simplified approach}. +It was introduced by Lieb in 1963\-~\cite{Li63}, but after a short series of papers by Lieb along with Sakakura\-~\cite{LS64} and Liniger\-~\cite{LL64}, it has largely been forgotten. +We have recently had a fresh look at this approach, in a collaboration with Carlen and Lieb\-~\cite{CJL20,CJL21,CHe21,Ja22,Ja23,Ja24} and found that its predictions exceeded our expectations. +Indeed, we found that it is as good as Bogolyubov theory at low densities\-~\cite{CJL20,CJL21} (at least as far as the ground state and condensate fraction are concerned), and it also reproduces the behavior of the Bose gas at high densities\-~\cite{CJL20}. +We have even found very good agreement for some systems {\it over the entire range of densities}\-~\cite{CHe21,Ja23}. +Lieb's Simplified approach consists in replacing the original problem that involves an infinite number of particles with a non-linear, non-local equation for a single particle. This is a familiar approach in physics, where many-body problems are replaced with non-linear effective equations. -The Simplified Approach actually gives us several levels of approximation, and defines a family of equations that are more or less accurate, and correspondingly, harder or easier to solve. -Some can be studied analytically, whereas others are only really amenable to a numerical analysis. +However, in most other settings, these effective equations are only accurate in one parameter regime. +For instance Bogolyubov theory is accurate at low densities. +In contrast, Lieb's Simplified approach is asymptotically accurate at both low- and high-densities, and it does quite well at intermediate densities as well. +The approach actually gives us several levels of approximation, and defines a family of equations that are more or less accurate, and correspondingly, harder or easier to solve. +Some can be studied analytically, whereas others have only been treated numerically. \bigskip \indent -In conclusion, systems of many interacting Bosons are interesting: they exist, are physically non-trivial, and mathematically challenging. +In short, systems of many interacting Bosons are interesting: they exist in the real world, exhibit non-trivial physical behaviors, and are mathematically challenging to study. +Lieb's Simplified approach is an approximation for the ground state of interacting Boson systems that is surprisingly accurate across densities, while being much simpler to study than the many-body system. +In this book, we will describe the derivation of Lieb's Simplified approach, and prove some properties of its solutions. +But first, let us take a step back and discuss the concept of Bose-Einstein condensation in more depth, and introduce the tools of statistical mechanics on which our entire premise is based. \bigskip {\bf Outline}:\par \indent -We will start out this course with a discussion of Bose-Einstein condensation in non-interacting systems, see Section\-~\ref{sec:statmech}. +In Chapter\-~\ref{sec:BEC}, we discuss Bose-Einstein condensation in the simplest setting in which it can be proved: for non-interacting systems. The arguments presented there follow the original derivation by Bose and Einstein\-~\cite{Bo24,Ei24}. -The discussion will begin with a much abbreviated introduction to quantum statistical mechanics. -Next, we will present the Simplified Approach, see Section\-~\ref{sec:simplified}. -In that section, we will also present the conjectures on the Bose gas that the Simplified Approach was developed to address. -In doing so, we will give an overview of scattering theory, which underlies the intuition in the low-density regime. -Some of the more abstract mathematical results needed for this section have been deferred to Appendices\-~\ref{app:functional_analysis} and\-~\ref{app:harmonic}. -Finally, in Section\-~\ref{sec:open}, we discuss some open problems. -There are also some exercises at the very end of this document. +We will open that section with a much abbreviated introduction to quantum statistical mechanics, for the benefit of readers with little experience in statistical mechanics. +Next, in Chapter\-~\ref{sec:bose_results} we will give a brief historical overview of some of the conjectures about the Bose gas that are most relevant to the discussion in this book. +In doing so, we will give a brief overview of scattering theory, which underlies the intuitive understanding of the low-density regime. +In Chapter\-~\ref{sec:simplified_def}, we introduce Lieb's Simplified approach. +We will discuss the approximations that are done to arrive at the various equations of the approach (in particular the Simple equation), and motivate it by discussing the accuracy of some of its predictions. +In Chapter\-~\ref{sec:existence}, we prove the existence of solutions to the Simple equation, and in Chapter\-~\ref{sec:predictions}, we prove asymptotic formulas for the prediction of the Simple equation for the energy and condensate fraction, and state similar results for the two-point correlation function and the momentum distribution. +These result use some functional and harmonic analysis, and we state the theorems that are needed without proof in Appendices\-~\ref{app:functional_analysis}, \ref{app:harmonic}, and\-~\ref{app:Ke}. +In Chapter\-~\ref{sec:numerics}, we briefly discuss the numerical computation of solutions to the Big and Medium equations. +Finally, in Chapter\-~\ref{sec:open}, we discuss some open problems. +\bigskip - -\section{Bose-Einstein condensation}\label{sec:statmech} +{\bf Acknowledgements}:\par \indent -Bose-Einstein condensation serves as the main motivation for the discussion in this course. -Proving the existence of Bose-Einstein condensation in interacting systems is the main open problem in the field, and the principle reason for the entire discussion that follows this section is to try new ways of approaching that. -In this section, we will discuss what Bose-Einstein condensation is in more detail, and prove that it occurs in a much simplified model: non-interacting Bosons. +This book is adapted from a course I gave at Texas State University, in the summer of 2023, organized by Jake Fillman, to whom I owe much credit! +The results described in this book are the fruit of a productive and very enjoyable collaboration with Eric A. Carlen and Elliott H. Lieb. +This book is a pedagogical summary of some of the results that we have discovered and published together (see the references below). + +\indent +I wholeheartedly thank Elliott H. Lieb for introducing me to his approach, and for very useful notes and comments on this manuscript. +I am also extremely grateful for his mentorship and support, and for our many enlightening discussions over the years. +I also thank Eric A. Carlen for many useful discussions, Markus Holzmann for carrying out the Quantum Monte Carlo computations shown in this book, as well as Erik Bahnson who read an early draft of this manuscript and gave very valuable feedback. +This work was partially supported by the Simons Foundation grant number 825876, and the National Science Foundation grant number DMS-2349077. + + + + +\section{Bose-Einstein condensation}\label{sec:BEC} +\indent +Bose-Einstein condensation serves as the main motivation for Lieb's Simplified approach. +Proving the existence of Bose-Einstein condensation in interacting systems is one of the main open problems in the field. +In this chapter, we will discuss what Bose-Einstein condensation is in more detail, and prove that it occurs in a much simplified model: non-interacting Bosons. But first, we must define what a ``Boson'' is, as well as the model we will be using. \bigskip @@ -163,14 +228,14 @@ But first, we must define what a ``Boson'' is, as well as the model we will be u \indent One of the great achievements of modern science is the understanding that macroscopic matter is made of microscopic elementary particles, and that the properties of the whole stem from the interaction between these elementary particles. The goal of statistical mechanics is to understand how this works, that is, the relation between the microscopic and the macroscopic worlds. +In order to motivate the main ideas of quantum statistical mechanics, let us begin with a short primer on classical systems. \bigskip \subsubsection{Classical statistical mechanics} \indent -Let us think classically for the moment. Classical particles are governed by Newton's second law of motion: $F=ma$. In the case of conservative systems (at their deepest level, all systems are actually conservative: friction is an effective description of a complicated, conservative behavior), this can be restated in the Hamiltonian formalism. -If I have $N$ particles, the Hamiltonian is a function on a $6N$-dimensional manifold (think $\mathbb R^{6N}$ for simplicity) that may look something like this: +A system of $N$ particles is described by a Hamiltonian, which is a function on $\mathbb R^{6N}$ that may look something like this: \begin{equation} H_N(x_1,\cdots,x_n;p_1,\cdots,p_N)=\sum_{i=1}^N\frac1{2m}p_i^2+\sum_{1\leqslant i0$ (yes, in this formalism, temperature is a parameter) and the {\it chemical potential} $\mu\in\mathbb R$. +In this setting, there are two parameters: the {\it temperature} $T>0$ (in this formalism, temperature is a parameter) and the {\it chemical potential} $\mu\in\mathbb R$. Given these two, the probability of a given configuration $(x_1,\cdots,x_N;p_1,\cdots,p_N)$ with $N$ particles is \begin{equation} \frac1{\Xi}e^{-\beta H_N(x_1,\cdots,x_N;p_1,\cdots,p_N)+\beta\mu N} @@ -205,17 +270,17 @@ in which $k_B$ is the {\it Boltzmann constant} and $\Xi$ is a normalization: . \end{equation} We immediately notice an issue: if the $x_i$ take values anywhere in $\mathbb R^3$, $\Xi$ will come out infinite! -To avoid this, let us restrict $x_i\in[0,L]^3$ for some finite $L$, and, later one, we will take $L\to\infty$. +To avoid this, let us restrict $x_i\in[0,L]^3$ for some finite $L$, and, later on, we will take $L\to\infty$. Note that $\Xi$ involves a sum over $N$: in the grand-canonical picture, we consider all possible numbers of particles. -In addition, note the $1/N!$ factor, which indicates that all the particles are identical, so the labelling $i=1,\cdots,N$ should not affect the statistics! +In addition, note the $1/N!$ factor, which indicates that all the particles are identical, and the labelling $i=1,\cdots,N$ does not affect the statistics. \bigskip \indent -Why do we choose a probability of the form\-~(\ref{gc})? -The real reason is that it is compatible with the laws of thermodynamics\-~\cite{Ru99,Ga99}, but that is a discussion for another time. -Instead, let us have a look at the properties of this distribution, and check that they make sense. -The first term in the exponential is $e^{-\beta H}$, and, since $\beta>0$, it says that configurations with large energies will be less likely. -The parameter $\beta$ (the temperature) quantifies how much less likely. +The precise reason why the probability is taken to be\-~(\ref{gc}) is not so simple. +There are several approaches to justifying it: by arguments based on finding {\it typical} configuration (see, for instance, \cite{GHe17} and references therein), or simply by arguing that this is the only way to reproduce the laws of thermodynamics (see, for instance,\cite{Ru99,Ga99}). +We will not enter in a profound discussion about this here, but instead, let us have check that this distribution is plausible, using some physical intuition. +The first term in the exponential is $e^{-\beta H}$, and, since $\beta>0$, it says that configurations with small energies will be more likely than those with large energies. +The parameter $\beta$ (the inverse of the temperature) quantifies how much more likely these are. This is rather sensible: we know that energy tends to be minimized, and that this is more true at low temperatures than higher ones. The second term is $e^{\beta\mu N}$ controls the number of particles. If $\mu>0$, then configurations with more particles are more likely, whereas $\mu<0$ makes more particles unlikely. @@ -226,9 +291,9 @@ Thus the chemical potential $\mu$ controls the {\it density} of particles. Why do we use the chemical potential instead of simply fixing the number of particles (which is, after all, the most direct way of setting the density)? This can be done: in statistical mechanics, the situation in which the number of particles is fixed is called the {\it canonical formalism}. The parameters are then the temperature and the number of particles. -One can show that the two formalisms are equivalent: there exists a mapping between the density and chemical potential that makes all average observables identical in both formalisms. +One can show that the two formalisms are equivalent: there exists a mapping between the density and chemical potential that makes all average observables identical in both formalisms\-~\cite{Ru99}. However, in certain situations, it may be easier to compute these averages in one formalism rather than the other. -In the case of Bose-Einstein condensation, the computation is easy in the grand-canonical, and impossibly difficult in the canonical, which is why we use the chemical potential instead of the density. +In the case of Bose-Einstein condensation, the computation is easy in the grand-canonical, and very difficult in the canonical, which is why we use the chemical potential instead of the density. \bigskip \indent @@ -268,7 +333,7 @@ The microscopic state of a system of quantum particles is given by a wavefunctio H_N=-\sum_{i=1}^N\frac1{2m}\Delta_{i}+\sum_{1\leqslant i =\lim_{L\to\infty}\frac1{L^3}\frac1{e^{\beta(\frac{q^2}{2m}-\mu_L)}-1} - 0 + =0 . \end{equation} Thus, choosing $\mu_L$ in this way, we find that the number of particles in the state $q=0$ is $L^3(\rho-\rho_c)$, that is, the number of particles in this state is proportional to $N$. @@ -592,29 +656,29 @@ This phenomenon is called Bose-Einstein condensation. \end{figure} \bigskip -\indent -This is significant for several reasons. -For one, this is a new phase of matter that is uniquely quantum: classical systems do not behave like this. -In addition, Bose-Einstein condensation may be the underlying mechanism for some truly astonishing phenomena: superfluidity and superconductivity. -Fluids flow, and, while doing so, they lose energy to friction. -This is visible in everyday life through the concept of {\it viscosity}. -A superfluid has zero viscosity, which means that it can flow without any friction. -This leads to truly astonishing behaviors: flows can separate and recombine without any losses, which allows superfluids to flow through membranes that are completely water-tight for ordinary fluids. -In addition, because no energy is lost through viscosity, the capillary force becomes dominant, and superfluids can be seen to creep up the sides of any container they are put in and escape. -This is fun and non-trivial behavior, but it also comes with serious technological applications. -The most astonishing of which may be superconductivity. -Electrical currents are caused by electrons flowing through metals. -As they do so, they are slowed by resistance, and the metal heats up. -Superconductivity is a phenomenon by which electrons can flow through a metal {\it without resistance}. -In most applications, resistance is waste, which makes superconducting metals extremely useful in many applications. -One way to understand the phenomenon is that electrons pair up into an effective excitation called a {\it Cooper pair}. -Now an electron is a Fermion, but a pair of electrons is a Boson, so Cooper pairs are Bosonic. -If these form the analogue of a superfluid phase, they can move around without resistance. -But what does all this have to do with Bose-Einstein condensation? -The answer is not so straightforward, but it has been posited that Bose-Einstein condensation is the underlying mechanism for both superfluidity and superconductivity. -This can be understood intuitively: if most particles are in the same state, they are all doing the same thing, and since friction comes from colliding particles, friction can disappear. -(This is a very handwavy explanation: the actual relation between Bose-Einstein condensation, superfluidity and superconductivity is a rather difficult subject). -\bigskip +%\indent +%This is significant for several reasons. +%For one, this is a new phase of matter that is uniquely quantum: classical systems do not behave like this. +%In addition, Bose-Einstein condensation may be the underlying mechanism for some truly astonishing phenomena: superfluidity and superconductivity. +%Fluids flow, and, while doing so, they lose energy to friction. +%This is visible in everyday life through the concept of {\it viscosity}. +%A superfluid has zero viscosity, which means that it can flow without any friction. +%This leads to truly astonishing behaviors: flows can separate and recombine without any losses, which allows superfluids to flow through membranes that are completely water-tight for ordinary fluids. +%In addition, because no energy is lost through viscosity, the capillary force becomes dominant, and superfluids can be seen to creep up the sides of any container they are put in and escape. +%This is fun and non-trivial behavior, but it also comes with serious technological applications. +%The most astonishing of which may be superconductivity. +%Electrical currents are caused by electrons flowing through metals. +%As they do so, they are slowed by resistance, and the metal heats up. +%Superconductivity is a phenomenon by which electrons can flow through a metal {\it without resistance}. +%In most applications, resistance is waste, which makes superconducting metals extremely useful in many applications. +%One way to understand the phenomenon is that electrons pair up into an effective excitation called a {\it Cooper pair}. +%Now an electron is a Fermion, but a pair of electrons is a Boson, so Cooper pairs are Bosonic. +%If these form the analogue of a superfluid phase, they can move around without resistance. +%But what does all this have to do with Bose-Einstein condensation? +%The answer is not so straightforward, but it has been posited that Bose-Einstein condensation is the underlying mechanism for both superfluidity and superconductivity. +%This can be understood intuitively: if most particles are in the same state, they are all doing the same thing, and since friction comes from colliding particles, friction can disappear. +%(This is a very handwavy explanation: the actual relation between Bose-Einstein condensation, superfluidity and superconductivity is a rather difficult subject). +%\bigskip \indent Bose-Einstein condensation was first predicted by Bose\-~\cite{Bo24} and Einstein\-~\cite{Ei24} in the 1920's (following roughly the argument given above). @@ -623,20 +687,20 @@ The experiments consist in trapping very cold Bosonic atoms (rubidium and sodium The trap is then released, and the momentum of the atoms measured. The result is a sharply peaked distribution: many atoms were in the same state! -\section{The Simplified Approach}\label{sec:simplified} -\subsection{Interacting Bose gasses} +\section{Knowns and unknowns of the interacting Bose gas}\label{sec:bose_results} +\subsection{Interacting Bose gases} \indent We have proved that the ideal Bose gas exhibits Bose-Einstein condensation. The approach we took consisted in computing exact expressions for the partition function $\Xi$ and the average occupation numbers $\left<\mathcal B_q\right>$. We were able to do this because the ideal Bose gas is an ``exactly solvable model'', but this is a very special feature of that specific model. -Furthermore, the ideal gas is not very realistic: there are no interactions between particles, which is not how real matter behaves. +Furthermore, the ideal gas is not very realistic: in that model, there are no interactions between particles, which is not how real matter behaves. Let us consider a more realistic Hamiltonian: \begin{equation} H_N=-\sum_{i=1}^N\frac1{2m}\Delta_{i}+\sum_{1\leqslant iR$, +so $\Delta\psi=0$ yields the solution, for $r>R$, \begin{equation} \psi=c-\frac ar \end{equation} @@ -867,15 +927,15 @@ This lemma gives us an alternative definition of the scattering length, which do \subsection{The Lee-Huang-Yang predictions}\label{sec:lhy} \indent By following the prescriptions of Bogolyubov theory, Lee, Huang and Yang\-~\cite{LHY57} made a number of predictions about the ground state (zero-temperature state) of the interacting Bose gas. -Here, we will focus on their predictions for the energy per particle, the condensate fraction and the two-point correlation function. +Here, we will focus on their predictions for the energy per particle and the condensate fraction. \bigskip \indent As was mentioned above, we study the ground state of the Hamiltonian\-~(\ref{Ham}). Let us be more specific. -As we did in section\-~\ref{sec:statmech}, we will consider the system to be in a finite volume with periodic boundary conditions. +As we did in Chapter\-~\ref{sec:BEC}, we will consider the system to be in a finite volume with periodic boundary conditions. We denote the volume by $V$, and place the particles on the torus $\mathbb R^3/(V^{\frac13}\mathbb Z)^3$. -Unlike section\-~\ref{sec:statmech}, we will work in the canonical ensemble, which means that we fix the number of particles to some $N\in\mathbb N$. +Unlike Chapter\-~\ref{sec:BEC}, we will work in the canonical ensemble, which means that we fix the number of particles to some $N\in\mathbb N$. We will be interested in the {\it thermodynamic limit}, in which \begin{equation} N\to\infty @@ -905,8 +965,8 @@ The ground-state energy per particle is defined as Lee, Huang and Yang predicted the following low-density expansion for the ground state energy. \bigskip -\theo{Conjecture}{\cite[(25)]{LHY57}}\label{conjecture:lhy_e} - For any potential $v\in L_1(\mathbb R^3)$ with $v\geqslant 0$ (actually, the potential may includes a hard-core component), as $\rho\to 0$, +\theoname{Conjecture}{{\rm\cite[(25)]{LHY57}}}\label{conjecture:lhy_e} + For any potential $v\in L_1(\mathbb R^3)$ with $v\geqslant 0$ (actually, the potential may include a hard-core component), as $\rho\to 0$, \begin{equation} e_0=\frac{2\pi}m\rho a\left(1+\frac{128}{15\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt{\rho})\right) \label{lhy_e} @@ -957,8 +1017,8 @@ The condensate fraction is then The Lee-Huang-Yang prediction of the condensation fraction for low densities is the following. \bigskip -\theoname{Conjecture}{\cite[(41)]{LHY57}}\label{conjecture:lhy_h} - For any potential $v\in L_1(\mathbb R^3)$ with $v\geqslant 0$ (actually, the potential may includes a hard-core component), as $\rho\to 0$, +\theoname{Conjecture}{{\rm\cite[(41)]{LHY57}}}\label{conjecture:lhy_h} + For any potential $v\in L_1(\mathbb R^3)$ with $v\geqslant 0$ (actually, the potential may include a hard-core component), as $\rho\to 0$, \begin{equation} 1-\eta_0\sim\frac{8\sqrt{\rho a^3}}{3\sqrt\pi} \label{lhy_h} @@ -975,71 +1035,79 @@ In particular, this conjecture would imply Bose-Einstein condensation, which mea \bigskip \indent -The two-point correlation function is the joint probability (in the usual quantum mechanical probability distribution $|\psi|^2$ of finding a particle at $y$ and another at $y'$: -\begin{equation} - C_2(y-y'):= - \lim_{\displaystyle\mathop{\scriptstyle N,V\to\infty}_{\frac NV=\rho}} - \sum_{i,j=1}^N\left<\psi_0\right|\delta(x_i-y)\delta(x_j-y')\left|\psi_0\right> - . - \label{C2} -\end{equation} -Lee, Huang and Yang made the following prediction. +Only the first of these two conjectures has been proved, under some mild assumptions on $v$: \bigskip -\theoname{Conjecture}{\cite[(48)]{LHY57}}\label{conjecture:lhy_C2} - For any potential $v\in L_1(\mathbb R^3)$ with $v\geqslant 0$ (actually, the potential may includes a hard-core component), as $\sqrt{\rho a}|x|\to \infty$, +\theoname{Theorem}{{\rm\cite{YY09,FS20,BCS21}}}\label{theo:lhy_e} + For any compactly supported potential $v\in L_3(\mathbb R^3)$ with $v\geqslant 0$, as $\rho\to 0$, \begin{equation} - \frac1{\rho^2}C_2(x)-1\sim\frac{16\rho a^3}{\pi^3(\sqrt{\rho a}|x|)^4} + e_0=\frac{2\pi}m\rho a\left(1+\frac{128}{15\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt{\rho})\right) + \label{lhy} + \end{equation} + where $a$ is the scattering length of the potential, and + \nopagebreakaftereq + \begin{equation} + \lim_{\rho\to 0}\frac{o(\sqrt{\rho})}{\sqrt{\rho}}=0 + . \end{equation} - where $a$ is the scattering length of the potential. \endtheo +\restorepagebreakaftereq \bigskip -\indent -Of these three conjectures, the only one that has been proved is Conjecture\-~\ref{conjecture:lhy_e}, and even so, the proof does not apply to all interactions $v$. -The first order term, $2\pi\rho a/m$ was proved by Lieb and Yngvason\-~\cite{LY98}, and the second order term was proved for potentials that are $L_1$ and $L_3$ by\-~\cite{FS20,FS23} and\-~\cite{YY09,BCS21}. -\cite{FS20} proved that\-~(\ref{lhy_e}) holds as an upper bound for soft-core potentials of finite range, and extended thir result to hard core potentials in\-~\cite{FS23}. -\cite{YY09}\-~proved the corresponding upper bound for small potentials, which was generalized to $L_3$ finite-range potentials by\-~\cite{BCS21}. -The other two conjectures are still wide open. + +The first order term, $2\pi\rho a/m$ was proved by Lieb and Yngvason\-~\cite{LY98}, and the second order term was proved by\-~\cite{FS20} and\-~\cite{YY09,BCS21}. +\cite{FS20} proved that\-~(\ref{lhy_e}) holds as a lower bound for soft-core potentials of finite range (they also extended their result to hard core potentials in\-~\cite{FS23}). +\cite{YY09}\-~proved the corresponding upper bound for small potentials, which was generalized to $L_3$ finite-range potentials by\-~\cite{BCS21} (the upper bound in the case of a hard-core potential is still open). +Conjecture\-~\ref{conjecture:lhy_h} is still wide open. In particular, there is still no proof of Bose-Einstein condensation for interacting models. -\subsection{Overview of the Simplified Approach} +\section{Definition of Lieb's Simplified approach}\label{sec:simplified_def} \indent -The Simplified Approach is what its name suggests: a simplified way of studying the interacting Bose gas that is considerably simpler than dealing with the Hamiltonian\-~(\ref{Ham}). +Proving properties of interacting Bose gases is a challenging task, and, so far, some major and natural questions are still open, such as proving Bose-Einstein condensation. +In this Chapter, we will introduce Lieb's Simplified approach, which is an approximation scheme that reproduces some of the physical behavior of the interacting Bose gas, while being much easier to study. +It is worth stressing that, at least as far as we understand, Lieb's Simplified approach is quite different from Bogolyubov theory, or from the rigorous methods that lead to\-~\cite{FS20}. +At the same time, the evidence discussed below shows that this approach reproduces several physical predictions for interacting Bose gases. +This makes it a promising candidate to study the mathematics and physics of interacting Bose gases. +\bigskip + +\subsection{Overview} +\indent +Lieb's Simplified approach is what its name suggests: a simplified way of studying the interacting Bose gas that is considerably simpler than dealing with the Hamiltonian\-~(\ref{Ham}). By using the term ``simplified'', we mean to imply that the results found using the approach are not (at least not yet) proven to be valid for the original Hamiltonian\-~(\ref{Ham}). In other words, it has (so far) not been used to prove any properties of\-~(\ref{Ham}) (though there is ongoing work in that direction). -However, we have ample numerical and analytical evidence that the Simplified Approach captures a lot of the physics of the Bose gas. -In fact, it captures significantly more than Bogolyubov theory, which is also an approximate approach. +However, we have ample numerical and analytical evidence that the Simplified approach captures a lot of the physics of the Bose gas. +In fact, it captures significantly more than Bogolyubov theory. This makes it quite an intriguing theory, and seems to be worth studying in more detail. -We will define the Simplified Approach in the next section; for now, let us discuss the results predicted by Simplified Approach, to motivate the more detailed study of it that will follow. +We will define Lieb's Simplified approach in the next section; for now, let us discuss the results predicted by the approach, to motivate the more detailed study that will follow. \bigskip \indent -The Simplified approach reduces the computation of the ground state energy to solving partial differential equations on $\mathbb R^3$ which is non-linear and non-local. -``Equation{\it s}'' plural, because the Simplified Approach actually provides a family of equations that are more or less accurate and, correspondingly, harder or easier to study. +Lieb's Simplified approach reduces the computation of the ground state energy to solving a partial differential equation on $\mathbb R^3$ which is non-linear and non-local. +Actually, it contains a family of equations. +Some are more accurate and harder to study, while others are easier to study but less accurate. Here we will discuss three of these equations, called the Big\-~(\ref{bigeq}), Medium\-~(\ref{medeq}) and Simple\-~(\ref{simpleq}) Equations. -Solving these equations is not easy, but it is much more approachable than solving the equation\-~(\ref{eigenvalue}), which involves an infinite number of particles, whereas the Simplified Approach provides single-particle equations. +Solving these equations is not, strictly speaking, easy, but it is much more so than solving the equation\-~(\ref{eigenvalue}), which involves an infinite number of particles, whereas Lieb's Simplified approach provides single-particle equations. \bigskip \indent -Within the Simplified approach, we have proved Conjectures\-~\ref{conjecture:lhy_e}\-~\cite{CJL20}, \ref{conjecture:lhy_h}\-~\cite{CJL21} and\-~\ref{conjecture:lhy_C2}\-~\cite{Ja22}. -In particular, we have proved that Bose-Einstein condensation occurs in the Simplified Approach. -Thus, the Simplified Approach tells us at least as much as Bogolyubov theory (at least for the ground state of\-~(\ref{Ham}): it does not have anything to say about the thermal state (at least for now)). -It goes quite a bit further. -The large density expansion of the ground state energy per particle has long been computed for the Bose gas, at least in the case in which the Fourier transform of the potential is non-negative: $\hat v\geqslant 0$: +Within the Simplified approach, we have proved analogous statements to Theorem\-~\ref{theo:lhy_e}\-~\cite{CJL20}, and Conjecture\-~\ref{conjecture:lhy_h}\-~\cite{CJL21}. +In particular, we have proved that Bose-Einstein condensation occurs in Lieb's Simplified approach. +Thus, it tells us at least as much as Bogolyubov theory (at least for the ground state of\-~(\ref{Ham}): it does not have anything to say about the thermal state (at least for now, see Chapter\-~\ref{sec:open})). +But it actually goes quite a bit further. +The large density expansion of the ground state energy per particle for the interacting Bose gas had been proved to be\-~\cite{Li63}, at least in the case in which the Fourier transform of the potential is non-negative: $\hat v\geqslant 0$, \begin{equation} e_0\sim_{\rho\to\infty}\frac\rho2\int dx\ v(x) . \label{e_highrho} \end{equation} -Within the Simplified Approach, this also holds. -This is where things get significant: the Simplified Approach gives accurate results for {\it both low and high densities}. +Within the Simplified approach, this also holds. +This is the first indication that something rather significant is going on: Lieb's Simplified approach gives accurate results for {\it both low and high densities}. This is rather unusual: there are many effective theories in physics (Bogolyubov theory is one of them), but in almost all such cases, the effective theory works in just one parameter regime. -The Simplified Approach works in two opposite regimes. +Lieb's Simplified approach works in two opposite ranges of density. \bigskip \indent -A natural question that arises is: if the Simplified Approach is accurate for small and large densities, how well does it do for intermediate densities? +A natural question that arises is: if Lieb's Simplified approach is accurate for small and large densities, how well does it do for intermediate densities? The energy per particle is plotted in Figure\-~\ref{fig:energy}. The red crosses come from a Quantum Monte Carlo computation, which is the gold standard for quantum mechanical predictions (even though the theoretical control over the accuracy of this method is relatively incomplete; quantum mechanics is hard...). The gray dotted line is the Bogolyubov prediction, which we see is good for small densities, but quickly falls off the mark. @@ -1055,7 +1123,7 @@ On the other hand, the Big Equation is a partial differential equation that can \begin{figure} \hfil\includegraphics[width=8cm]{energy.pdf} \caption{% - \cite[Fig 1]{CHe21} The predictions of the energy per particle as a function of the density for the \eqformat{Simple Equation}, \eqformat{Medium Equation}, and \eqformat{Big Equation}, compared to a \eqformat{Quantum Monte Carlo} (QMC) simulation and the Lee-Huang-Yang (LHY) formula. + \cite[Fig 1]{CHe21} The predictions of the energy per particle as a function of the density for the \eqformat{Simple Equation}, \eqformat{Medium Equation}, and \eqformat{Big Equation}, compared to a \eqformat{Quantum Monte Carlo} (QMC) simulation (computed by M.\-~Holzmann) and the Lee-Huang-Yang (LHY) formula, for $v=e^{-|x|}$. } \label{fig:energy} \end{figure} @@ -1069,23 +1137,23 @@ The Big and Medium equations, on the other hand, seem to be very accurate throug \begin{figure} \hfil\includegraphics[width=8cm]{condensate.pdf} \caption{% - \cite[Fig 3]{CHe21} The predictions of the condensate fraction as a function of the density for the \eqformat{Simple Equation}, \eqformat{Medium Equation}, and \eqformat{Big Equation}, compared to a \eqformat{Quantum Monte Carlo} (QMC) simulation and the Bogolyubov prediction (Bog). + The predictions of the condensate fraction as a function of the density for the \eqformat{Simple Equation}, \eqformat{Medium Equation}, and \eqformat{Big Equation}, compared to a \eqformat{Quantum Monte Carlo} (QMC) simulation (computed by M.\-~Holzmann) and the Bogolyubov prediction (Bog), for $v=e^{-|x|}$. } \label{fig:condensate} \end{figure} \indent -Thus, the Simplified Approach provides us with a simple tool to study systems of interacting Bosons. -The simplest equations can be studied analytically, and reproduce known and conjectured results\-~\cite{CJL20,CJL21,Ja22}. -The more complicated equations are surprisingly accurate, while being much simpler to handle than traditional methods\-~\cite{CHe21,Ja23b}. -In particular, the Big Equation has been used\-~\cite{Ja23b} to find evidence for a previously undiscovered phase in Bosons at intermediate densities. +Thus, Lieb's Simplified approach provides us with a simple tool to study systems of interacting Bosons. +The simplest equation can be studied analytically, and reproduces known and conjectured results\-~\cite{CJL20,CJL21,Ja22}. +The more complicated equations are surprisingly accurate, while being much simpler to handle than traditional methods\-~\cite{CHe21,Ja23}. +In particular, the Big Equation has been used\-~\cite{Ja23} to find evidence for a previously unknown phase in Bosons at intermediate densities. -\subsection{The Simplified Approach and its approximations} +\subsection{Lieb's Simplified approach and its approximations}\label{sec:simplified_construction} \indent -The Simplified Approach dates back to a paper by Lieb from 1963\-~\cite{Li63}. +The Simplified approach dates back to a paper by Lieb from 1963\-~\cite{Li63}. The discussion below follows\-~\cite{Li63} closely (though with more detail). -The starting point of the Simplified Approach is\-~(\ref{eigenvalue}), which we write out using\-~(\ref{Ham}): +The starting point is\-~(\ref{eigenvalue}), which we write out using\-~(\ref{Ham}): \begin{equation} -\sum_{i=1}^N\frac1{2m}\Delta_{i}\psi_0+\sum_{1\leqslant i\max\{\frac d2,1\}$ and $v\geqslant 0$, then\-~(\ref{simpleq}) has an integrable solution $u$ satisfying $0\leqslant u(x)\leqslant 1$. \endtheo \bigskip +(In the rest of this book, we assume that $v$ is not identically zero, otherwise the solution or the Simple equation is $u=0$.) +\bigskip + \indent The crucial idea of the proof is that the Simple Equation is much easier to solve if we change the point of view: instead of fixing $\rho$ and computing $e$ and $u$, we fix $e$ and compute $\rho$ and $u$. Doing so turns the term $-4eu$ into a linear term. @@ -1414,6 +1490,7 @@ Next, we write the Simple Equation as a fixed point equation, by defining \left(-\frac1m\Delta+v+4e\right)u_n=v+2e\rho_{n-1} u_{n-1}\ast u_{n-1} ,\quad u_0:=0 + \label{un_def} \end{equation} with \begin{equation} @@ -1440,57 +1517,56 @@ But first, let us prove a simple useful lemma. 0=\frac{2e}\rho-4e\int dx\ u(x)+2e\rho\left(\int dx\ u(x)\right)^2 \end{equation} which we solve: + \nopagebreakaftereq \begin{equation} \int dx\ u(x)=\frac1\rho . \end{equation} \qed +\restorepagebreakaftereq \bigskip \indent -Let us translate this to the language of functional analysis: let +It is convenient to rewrite the iteration\-~(\ref{un_def}): let \begin{equation} K_e:=\left(-\frac1m\Delta+v+4e\right)^{-1} + \label{Ke} \end{equation} -which is an operator from the Sobolev space $W_{2,p}(\mathbb R^d)$ to $L_p(\mathbb R^d)$ (see Appendix\-~\ref{app:sobolev}). -Indeed, if $f\in W_{2,p}(\mathbb R^d)$, then, by the Sobolev inequality, $f\in L_q(\mathbb R^d)$ with $\frac1q=\frac1p-\frac 2d$ and, by the triangle and H\"older inequalities, -\begin{equation} - \left\|-\frac1m\Delta f+vf+4ef\right\|_p - \leqslant - \frac1m\|\Delta f\|_p+4e\|f\|_p+\|v\|_{\frac d2}\|f\|_q - <\infty -\end{equation} -so $(-\frac1m\Delta+v+4e)$ maps $L_2$ to $W_{2,p}$ (this is where the condition that $v$ be at least $L_{\frac d2}$ comes from). +which is a well-defined operator when $v\in L_p(\mathbb R^d)$ with $p>\frac d2$, see Appendix\-~\ref{app:Ke}. +As is proved in Lemma\-~\ref{lemma:Ke_pos}, $K_e$ is a bounded operator from $L_p(\mathbb R^d)$ to $W_{2,p}(\mathbb R^d)$ (see Appendix\-~\ref{app:sobolev}). +Actually, we prove in Lemma\-~\ref{lemma:Ke_extend}, that $K_e$ is also well defined on functions in $L_q(\mathbb R^d)$ for any $q\geqslant 1$. +In addition, the operator $K_e$ can be shown to be positivity preserving, see Lemma\-~\ref{lemma:Ke_pos}, that is, if $u\geqslant 0$, then $K_e u\geqslant 0$. In terms of $K_e$, \begin{equation} u_n=K_e(v+2e\rho_{n-1}u_{n-1}\ast u_{n-1}) . \end{equation} +The terms on the right side are well-defined whenever $v$ and $u_{n-1}$ are integrable (it follows from Young's inequality, Theorem\-~\ref{theo:young}, that $u_{n-1}\ast u_{n-1}$ is integrable if $u_{n-1}$ is). \bigskip -\theo{Lemma}\label{lemma:Ke_pos} - The operator $K_e$ is positivity preserving, see Appendix\-~\ref{app:positivity_preserving}. -\endtheo -\bigskip - -\indent\underline{Proof}: - This follows directly from Theorem\-~\ref{theo:schrodinger}, taking $v$ to be $m(v+4e)$ and the fact that $\mathrm{spec}(-\Delta+m(x+4e)\geqslant 4em$. -\qed -\bigskip \theo{Lemma}\label{lemma:monotone} $\rho_n$ and $u_n$ are pointwise increasing in $n$ and \nopagebreakaftereq \begin{equation} + 0\leqslant u_n(x)\leqslant 1 + ,\quad \int dx\ u_n(x)\leqslant\frac1{2e}\int dx\ (1-u_n(x))v(x)\equiv\frac1{\rho_n} . + \label{inductive_lem} \end{equation} \endtheo \restorepagebreakaftereq \bigskip \indent\underline{Proof}: - We proceed by induction. + We prove by induction on $n\geqslant 0$ that\-~(\ref{inductive_lem}) holds and that + \begin{equation} + u_{n+1}(x)\geqslant u_n(x) + ,\quad + \rho_{n+1}\geqslant \rho_n + . + \end{equation} \bigskip \point @@ -1500,40 +1576,48 @@ In terms of $K_e$, ,\quad \rho_0=\frac{2e}{\int dx\ v(x)} \end{equation} - and + so + \begin{equation} + 0\leqslant u_0\leqslant 1 + ,\quad + \int dx\ u_0(x)\leqslant\frac1{\rho_n} + . + \end{equation} + We are left with proving that $u_1\geqslant u_0$ and $\rho_1\geqslant \rho_0$. + We have \begin{equation} u_1=K_ev ,\quad \rho_1=\frac{2e}{\int dx\ v(x)-\int dx\ v(x)K_ev(x)} - . \end{equation} - By Lemma\-~\ref{lemma:Ke_pos}, + and since $K_e$ is positivity preserving, \begin{equation} u_1\geqslant 0\equiv u_0 ,\quad \rho_1\geqslant\frac{2e}{\int dx\ v(x)}\equiv \rho_0 . \end{equation} - Finally, $u_1$ satisfies - \begin{equation} - -\frac1m\Delta u_1+4e u_1+vu_1=v - \end{equation} - so, integrating both sices, - \begin{equation} - \int dx\ u_1(x)=\frac1{4e}\int dx\ v(x)(1-u_1(x)) - \leqslant\frac1{2e}\int dx\ v(x)(1-u_1(x)) - . - \label{intu1} - \end{equation} + + %Finally, $u_1$ satisfies + %\begin{equation} + % -\frac1m\Delta u_1+4e u_1+vu_1=v + %\end{equation} + %so, integrating both sides, + %\begin{equation} + % \int dx\ u_1(x)=\frac1{4e}\int dx\ v(x)(1-u_1(x)) + % \leqslant\frac1{2e}\int dx\ v(x)(1-u_1(x)) + % . + % \label{intu1} + %\end{equation} \bigskip \point - Suppose the lemma holds for $n\geqslant 1$. - By Lemma\-~\ref{lemma:Ke_pos} and the inductive hypothesis, + Suppose the lemma holds for $n\geqslant 0$. + Using the fact that $K_e$ is positivity preserving, \begin{equation} u_{n+1}=K_ev+2e\rho_nK_e u_n\ast u_n\geqslant K_ev+2e\rho_{n-1}K_e u_{n-1}\ast u_{n-1} - \equiv u_n + \equiv u_n\geqslant 0 . \end{equation} In addition, @@ -1542,9 +1626,9 @@ In terms of $K_e$, \geqslant\frac{2e}{\int dx\ v(x)(1-u_{n}(x))}=\rho_n . \end{equation} - Finally, + Finally, by\-~(\ref{un_def}), \begin{equation} - -\frac1m\Delta u_{n+1}+4eu_{n+1}+v u_{n+1}=v+2e\rho_n u_n\ast u_n + -\frac1m\Delta u_{n+1}+4eu_{n+1}=v(1-u_{n+1})+2e\rho_n u_n\ast u_n \end{equation} so, integrating both sides, \begin{equation} @@ -1578,40 +1662,10 @@ In terms of $K_e$, \leqslant\frac1{2e}\int dx\ v(x)(1-u_{n+1}(x)) . \end{equation} -\qed -\bigskip - -\theo{Lemma}\label{lemma:bound} - For all $n$, - \nopagebreakaftereq - \begin{equation} - 0\leqslant u_n(x)\leqslant 1 - . - \end{equation} -\endtheo -\restorepagebreakaftereq -\bigskip - -\indent\underline{Proof}: - We proceed by induction. - The lemma is trivial for $n=0$ since $u_0=0$. \bigskip \point - First, we prove that $u_{n+1}\geqslant 0$. - We have - \begin{equation} - u_{n+1}=K_e(v+2e\rho_n u_n\ast u_n) - \end{equation} - so, by Lemma\-~\ref{lemma:Ke_pos}, - \begin{equation} - u_{n+1}\geqslant 0 - . - \end{equation} - \bigskip - - \point - We now turn to $u_{n+1}\leqslant 1$. + Finally, we prove that $u_{n+1}\leqslant 1$. We have \begin{equation} \frac1m\Delta u_{n+1}=v(u_{n+1}-1)+4eu_{n+1}-2e\rho_n u_n\ast u_n @@ -1622,7 +1676,7 @@ In terms of $K_e$, |u_n\ast u_n|\leqslant\|u_n\|_1\|u_n\|_\infty \leqslant\|u_n\|_1 \end{equation} - and, by Lemma\-~\ref{lemma:monotone}, $\|u_n\|_1\leqslant\frac1{\rho_n}$ so + and, by the inductive hypothesis, $\|u_n\|_1\leqslant\frac1{\rho_n}$ so \begin{equation} \frac1m\Delta u_{n+1}\geqslant v(u_{n+1}-1)+4eu_{n+1}-2e @@ -1640,18 +1694,20 @@ In terms of $K_e$, \geqslant2e>0 \end{equation} Therefore, $u_{n+1}$ is subharmonic on $A$ (see Appendix\-~\ref{app:harmonic}). - By Theorem\-~\ref{theo:harmonic} $u_{n+1}$ achieves its maximum on the boundary of $A$, but, by definition, $u_{n+1}=1$ on its boundary. + By Theorem\-~\ref{theo:harmonic} $u_{n+1}$ achieves its maximum on the boundary of $A$, but, by definition, $u_{n+1}=1$ on its boundary ($u_{n+1}$ is continuous by\-~(\ref{un_def})). Therefore, inside $A$, \begin{equation} u_{n+1}\leqslant 1 \end{equation} which is a contradiction. - Therefore $A$ is empty (note that $A$ is an open set because $u_{n+1}$ is continuous, by virtue of being in $W_{2,p}(\mathbb R^d)$) and so + Therefore $A$ is empty and so + \nopagebreakaftereq \begin{equation} u_{n+1}\leqslant 1 . \end{equation} \qed +\restorepagebreakaftereq \bigskip \theo{Lemma} @@ -1667,7 +1723,7 @@ In terms of $K_e$, \bigskip \indent\underline{Proof}: - The limit of $u_n$ exist by Lemmas\-~\ref{lemma:monotone} and\-~(\ref{lemma:bound}) and the monotone convergence theorem. + The limit of $u_n$ exist by Lemma\-~\ref{lemma:monotone} and the monotone convergence theorem. Furthermore, by Lemma\-~\ref{lemma:monotone}, \begin{equation} \rho_n\leqslant\frac1{\int dx\ u_n(x)} @@ -1680,7 +1736,7 @@ In terms of $K_e$, \int dx\ u(x)\leqslant \frac1{2e}\|v\|_1 \end{equation} so $u\in L_1(\mathbb R^d)$, and so, by dominated convergence, $u_n$ converges in $L_1(\mathbb R^d)$. - Moreover, by Lemma\-~\ref{lemma:bound}, + Moreover, since $u\leqslant 1$, \begin{equation} \|u\|_p^p\leqslant\|u\|_1 \end{equation} @@ -1704,7 +1760,7 @@ In terms of $K_e$, +2e\|u_n\ast u_n\|_p(\rho-\rho_n) . \end{equation} - Thus, $v+2e\rho_n u_n\ast u_n$ converges in $L_p$, so $K_e(v+2e\rho_n u_n\ast u_n)$ converges in $W_{2,p}$. + Thus, $v+2e\rho_n u_n\ast u_n$ converges in $L_p$, so, since $K_e$ is bounded from $L_p$ to $W_{2,p}$ (see above), $K_e(v+2e\rho_n u_n\ast u_n)$ converges in $W_{2,p}$. Therefore, \begin{equation} u=K_e(v+2e\rho u\ast u) @@ -1758,7 +1814,7 @@ To prove Theorem\-~\ref{theo:existence}, we need to prove that $e\mapsto\rho(e)$ \frac1\rho+b_1-2a_1=\sum_{n=1}^\infty \rho_n(b_n-a_n)^2 . \end{equation} - Now, by\-~(\ref{intu1}), + Now, by\-~(\ref{intun}), \begin{equation} a_1=\frac1{4e}\int dx\ v(x)(1-u_1(x)) =\frac12b_1 @@ -1785,14 +1841,16 @@ To prove Theorem\-~\ref{theo:existence}, we need to prove that $e\mapsto\rho(e)$ \end{equation} so $\rho(0)=0$, and \begin{equation} - \lim_{e\to\infty}\rho(e)=\infty + \lim_{e\to\infty}\rho_n(e)=\infty \end{equation} so, since $\rho_n$ converges uniformly on all compacts, + \nopagebreakaftereq \begin{equation} \lim_{e\to\infty}\rho(e)=\infty . \end{equation} \qed +\restorepagebreakaftereq \bigskip \indent @@ -1841,37 +1899,48 @@ But what of the uniqueness? =\int dx\ u(x)=\frac1\rho \end{equation} so + \nopagebreakaftereq \begin{equation} u= u' . \end{equation} \qed +\restorepagebreakaftereq \bigskip \indent Thus, taking the point of view in which the energy $e$ is fixed and $\rho$ is computed as a fuction of $e$, the solution is unique. However, this does not imply that the solution to the problem in which $\rho$ is fixed is unique: the mapping $e\mapsto\rho(e)$ may not be injective (it is surjective by Lemma\-~\ref{lemma:surjective}). -To prove that it is injective, one could prove that $\rho$ is an increasing function of $e$ (physically, it should be: the higher the density is, the higher the energy should be because the potnetial is repulsive). -This has been proved for small and large values of $e$\-~\cite{CJL21}, but, in general, it is still an open problem. +To prove that it is injective, one could prove that $\rho$ is an increasing function of $e$ (physically, it should be: the higher the density is, the higher the energy should be because the potential is repulsive). +This has been proved for small and large values of $e$\-~\cite{CJL21}, but, in general, it is still an open problem, see Chapter\-~\ref{sec:open}. + +\section{Predictions of the Simple Equation}\label{sec:predictions} +\indent +Let us now discuss predictions of the Simple equation for the ground state energy and the condensate fraction. +As we will see, the prediction for the ground state energy is asymptotically correct at both low and high density. +These prediction for the condensate fraction coincides at low densities with that from Bogolyubov theory. +In this chapter, we will restrict our attention to the three-dimensional case. +Adapting these results to other dimensions is an open problem, see Chapter\-~\ref{sec:open}. +\bigskip \subsection{Energy of the Simple Equation} \indent -The Simplified Approach provides a natural prediction for the ground state energy per-particle (see\-~(\ref{E0})): +Lieb's Simplified approach provides a natural prediction for the ground state energy per-particle (see\-~(\ref{E0})): \begin{equation} e=\frac\rho2\int dx\ (1-u(x))v(x) . \end{equation} -We have already seen numerical evidence for the fact that the Big and Medium equations provide very accurate predictions for the energy, see Figure\-~\ref{fig:energy}. -Let us now discuss an analytical result, which we can prove for the Simple Equation. +We have already seen numerical evidence for the fact that the Big and Medium equations provide very accurate predictions for the energy at all densities, see Figure\-~\ref{fig:energy}. +The Simple equation may not be so accurate at intermediate, but we can prove that it predicts the correct energy at low and high densities. \bigskip \indent -As was explained in Section\-~\ref{sec:lhy}, the ground state energy of the Bose gas has been computed at low and high densities, see\-~(\ref{lhy_e}) and\-~(\ref{e_highrho}). -We have proved that both of these asymptotic expansions hold for the prediction of the Simple Equation. +As we discussed in Section\-~\ref{sec:lhy}, the ground state energy of the Bose gas has been computed at low and high densities, see Theorem\-~\ref{theo:lhy_e} and\-~(\ref{e_highrho}). +We have proved that the prediction of the Simple equations agrees with both of these asymptotic expansions. \bigskip -\theoname{Theorem}{\cite[Theorem 1.4]{CJL20}}\label{theo:energy} - For the Simple Equation, under the assumptions of Theorem\-~\ref{theo:existence} for $d=3$, as $\rho\to0$, +\theoname{Theorem}{{\rm\cite[Theorem 1.4]{CJL20}}}\label{theo:energy} + For the Simple Equation, for $d=3$, under the assumptions of Theorem\-~\ref{theo:existence}, as $\rho\to0$, \begin{equation} e=\frac{2\pi}m\rho a\left(1+\frac{128}{15\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt\rho)\right) \label{low_density} @@ -1916,7 +1985,7 @@ The asymptotic agreement at high densities therefore only holds for positive typ + \int_{\mathbb R^d\setminus\chi_a}dx\ u(x)v(x) \end{equation} - which, by Lemmas\-~\ref{lemma:intu} and\-~\ref{lemma:bound}, is bounded by + which, by Lemma\-~\ref{lemma:intu}, is bounded by \begin{equation} \int dx\ u(x)v(x) \leqslant @@ -1963,9 +2032,14 @@ The asymptotic agreement at high densities therefore only holds for positive typ e_w-\frac{2\pi}m\rho a = -\frac{16\sqrt 2(me)^{\frac32}}{15\pi^2}\int dx\ \varphi(x) v(x)+o(\rho^{\frac32}) - . \label{ewephi} \end{equation} + where $1-\varphi$ satisfies the scattering equation\-~(\ref{scateq}): + \begin{equation} + (-\frac1m\Delta+v)\varphi=v(x) + . + \label{scateq_varphi} + \end{equation} Summing\-~(\ref{eew}) and\-~(\ref{ewephi}), we find \begin{equation} e=\frac{2\pi}m\rho a\left(1+\frac{32\sqrt 2(me)^{\frac32}}{15\pi^2\rho}+o(\sqrt\rho)\right) @@ -2030,9 +2104,9 @@ The asymptotic agreement at high densities therefore only holds for positive typ s\mapsto\left|\tilde k^2+1-\sqrt{(\tilde k^2+1)^2-s}-\frac{s}{2\tilde k^2}\right| \end{equation} is monotone increasing. - In addition, by~\-(\ref{S}) and~\-(\ref{simpleq}), and using the fact that $u(x)\leqslant 1$ (see Lemma\-~\ref{lemma:bound}) and $v(x)\geqslant 0$, + In addition, by~\-(\ref{S}) and~\-(\ref{simpleq}), and using the fact that $u(x)\leqslant 1$ and $v(x)\geqslant 0$, \begin{equation} - |\hat S(k)|\leqslant\int dx\ |(1-u(x))v(x)|=\frac{2e}\rho + |\hat S(k)|\leqslant\int dx\ (1-u(x))v(x)=\frac{2e}\rho . \end{equation} Therefore @@ -2052,13 +2126,14 @@ The asymptotic agreement at high densities therefore only holds for positive typ \frac{(me)^{\frac32}}{\rho\pi^3}\int d\tilde k\ \left|\tilde k^2+1-\sqrt{(\tilde k^2+1)^2-1}-\frac{1}{2\tilde k^2}\right| =\frac{32\sqrt 2(me)^{\frac32}}{15\pi^2\rho} . + \label{bound_absuw} \end{equation} By dominated convergence, and using the fact that $\hat S(0)=\frac{2e}\rho$, \begin{equation} \begin{largearray} \lim_{e\to 0}\frac1{(me)^{\frac32}}(e-e_w) =-\lim_{e\to 0}\frac\rho{2(me)^{\frac32}}\int dx\ (u(x)-w(x))v(x) - \\[0.5cm]\hfill + =\\[0.5cm]\hfill = -\frac12\int dx\ v(x)\left( \frac1{\pi^3}\int\left(\tilde k^2+1-\sqrt{(\tilde k^2+1)^2-1}-\frac1{2\tilde k^2}\right)d \tilde k @@ -2075,17 +2150,23 @@ The asymptotic agreement at high densities therefore only holds for positive typ This proves\-~(\ref{eew}). Incidentally, again by dominated convergence, \begin{equation} - u(x)-w(x)= - \frac{(me)^{\frac32}}{\rho\pi^3}\int\left(\tilde k^2+1-\sqrt{(\tilde k^2+1)^2-1}-\frac{1}{2\tilde k^2}\right)\ d\tilde k - =-\frac{32\sqrt 2(me)^{\frac32}}{15\pi^2\rho}+\sqrt\rho f_\rho(x) - \label{uapproxw} + \begin{largearray} + u(x)-w(x)= + \frac{(me)^{\frac32}}{\rho\pi^3}\int\left(\tilde k^2+1-\sqrt{(\tilde k^2+1)^2-1}-\frac{1}{2\tilde k^2}\right)\ d\tilde k + +\sqrt\rho f_\rho(x) + =\\[0.5cm]\hfill + =-\frac{32\sqrt 2(me)^{\frac32}}{15\pi^2\rho}+\sqrt\rho f_\rho(x) + \label{uapproxw} + \end{largearray} \end{equation} with \begin{equation} - 0\leqslant f_\rho(x)\leqslant\frac{32\sqrt2(me)^{\frac32}}{15\pi^2\rho} - ,\quad f_\rho(x)\mathop{\longrightarrow}_{\rho\to0}0 \end{equation} + and by\-~(\ref{bound_absuw}), + \begin{equation} + 0\leqslant f_\rho(x)\leqslant\frac{64\sqrt2(me)^{\frac32}}{15\pi^2\rho^{\frac32}}=O(1) + \end{equation} pointwise in $x$. \bigskip @@ -2096,7 +2177,7 @@ The asymptotic agreement at high densities therefore only holds for positive typ \xi(r):=w(r)-\varphi(r). \label{xi} \end{equation} - By\-~(\ref{u1}), (\ref{scateq}) and\-~(\ref{simpleq}), + By\-~(\ref{u1}), (\ref{scateq_varphi}) and\-~(\ref{simpleq}), \begin{equation} (-\Delta+v(x))\xi(x)=-(u(x)-w(x))v(x) . @@ -2104,13 +2185,13 @@ The asymptotic agreement at high densities therefore only holds for positive typ Therefore, by Lemma\-~\ref{lemma:scattering}, \begin{equation} e_w-\frac{2\pi}m\rho a=-\frac\rho2\int dx\ \xi(x)v(x) - =-\frac\rho2\int dx\ v(x)(-\Delta+v)^{-1}((u-w)v)(x) \end{equation} - and + and so, by\-~(\ref{scateq_varphi}), \begin{equation} - (-\Delta+v)^{-1}v(x)=\varphi(x) + e_w-\frac{2\pi}m\rho a + =-\frac\rho2\int dx\ \xi(x)(-\Delta+v)\varphi(x) \end{equation} - so + and so, since $-\Delta +v$ is symmetric, \begin{equation} e_w-\frac{2\pi}m\rho a =-\frac\rho2\int dx\ \varphi(x)(u(x)-w(x))v(x) @@ -2126,20 +2207,93 @@ The asymptotic agreement at high densities therefore only holds for positive typ Since $x\mapsto f_\rho(x)$ is bounded, we can use dominated convergence to show\-~(\ref{ewephi}). \qed -\subsection{Condensate fraction} +\subsection{Condensate fraction}\label{sec:condensate_fraction} \indent -The computation of the condensate fraction is not a sstraightforward as the energy. -In fact, the Simplified Approach really only gives a clear prediction for the energy, through\-~(\ref{E0}) which translates into\-~(\ref{energy}). +The computation of the condensate fraction is not a straightforward as the energy. +In fact, Lieb's Simplified approach really only gives a clear prediction for the energy, through\-~(\ref{E0}) which translates into\-~(\ref{energy}). How do we compute the condensate fraction without knowing the expression of the wavefunction? -The idea is to rewrite the condensate fraction in terms of the ground-state energy of an effective Hamiltonian. -This is done in detail in Exercise\-~\ref{ex:feynman_hellman}. -Proceeding in this way, we get a prediction for the condensate fraction. -As was shown in Figure\-~\ref{fig:condensate}, the prediction of the Big and Medium equations are rather accurate for all densities, as the energy was. -For the Simple Equation, we can prove an asymptotic expansion for low densities. +The idea is to rewrite the condensate fraction in terms of the ground-state energy of an effective Hamiltonian: +\begin{equation} + \tilde H_N(\epsilon):= + H_N+\epsilon\sum_{i=1}^NP_i +\end{equation} +where $P_i$ is the projector onto the constant state\-~(\ref{Pi}) (recall\-~(\ref{eta_Pi})). +Let $\tilde E_0(\epsilon)$ denote the ground state energy of $\tilde H_N(\epsilon)$, and $\tilde\psi_0(\epsilon)$ denote the ground state of $\tilde H_N(\epsilon)$ with $\|\tilde\psi_0(\epsilon)\|_2=1$. +We have +\begin{equation} + \tilde E_0(\epsilon) + =\left<\tilde\psi_0(\epsilon)\right|\tilde H_N(\epsilon)\left|\tilde\psi_0(\epsilon)\right> +\end{equation} +so +\begin{equation} + \partial_\epsilon\tilde E_0|_{\epsilon=0} + = + 2\left<\partial_\epsilon\tilde\psi_0|_{\epsilon=0}\right|H_N\left|\psi_0\right> + + + \left<\psi_0\right|\partial_\epsilon\tilde H_N|_{\epsilon=0}\left|\psi_0\right> +\end{equation} +that is +\begin{equation} + \partial_\epsilon\tilde E_0|_{\epsilon=0} + = + 2E_0\left<\partial_\epsilon\tilde\psi_0|_{\epsilon=0}|\psi_0\right> + + + \left<\psi_0\right|\sum_{i=1}^NP_i\left|\psi_0\right> + . +\end{equation} +In addition, +\begin{equation} + 2\left<\partial_\epsilon\tilde\psi_0|_{\epsilon=0}|\psi_0\right> + = + \partial_\epsilon\left<\tilde\psi_0(\epsilon)|\tilde\psi_0(\epsilon)\right>|_{\epsilon=0} + = + \partial_\epsilon1|_{\epsilon=0} + =0 + . +\end{equation} +Thus, +\begin{equation} + \frac1N + \partial_\epsilon\tilde E_0|_{\epsilon=0} + = + \frac1N\sum_{i=1}^N\left<\psi_0\right|P_i\left|\psi_0\right> + . +\end{equation} +Therefore, +\begin{equation} + \eta_0= + \lim_{\displaystyle\mathop{\scriptstyle N,V\to\infty}_{\frac NV=\rho}} + \frac1N + \partial_\epsilon\tilde E_0|_{\epsilon=0} + . + \label{condensate_feynman_hellman} +\end{equation} \bigskip -\theoname{Theorem}{\cite[Theorem 1.6]{CJL21}} - For the Simple Equation, if $(1+|x|^4)v(x)\in L_1(\mathbb R^3)\cap L_2(\mathbb R^3)$ and $v\geqslant 0$, as $\rho\to0$, +\indent +This construction gives us a natural prediction of the condensate fraction for the Simple equation. +To define it, we repeat the arguments of Sections\-~\ref{sec:simplified_construction}-\ref{sec:equations} and Appendix\-~\ref{app:proof_factorization} to the modified Hamiltonian. +The corresponding modified Simple equation is\-~\cite{CJL21}: +\begin{equation} + (-\Delta+2\epsilon+4e)u_\epsilon=(1-u_\epsilon)v+2\rho e_\epsilon u_\epsilon\ast u_\epsilon + ,\quad + e_\epsilon=\frac\rho2\int dx\ (1-u_\epsilon(x)) + \label{simpleq_eta} +\end{equation} +from which we define the prediction of the condensate fraction by analogy with\-~(\ref{condensate_feynman_hellman}): +\begin{equation} + \eta:=\partial_\epsilon e_\epsilon|_{\epsilon=0} + . +\end{equation} +\bigskip + +\indent +As was shown in Figure\-~\ref{fig:condensate}, the prediction of the Big and Medium equations are rather accurate for all densities, as the energy was. +For the Simple Equation, we can prove an asymptotic expansion for low densities, which agrees with the prediction of Bogolyubov theory in Conjecture\-~\ref{conjecture:lhy_h}. +\bigskip + +\theoname{Theorem}{{\rm\cite[Theorem 1.6]{CJL21}}} + For the Simple Equation, in $d=3$, if $(1+|x|^4)v(x)\in L_1(\mathbb R^3)\cap L_2(\mathbb R^3)$ and $v\geqslant 0$, as $\rho\to0$, \begin{equation} 1-\eta\sim\frac{8\sqrt{\rho a^3}}{3\sqrt\pi} \end{equation} @@ -2147,46 +2301,881 @@ For the Simple Equation, we can prove an asymptotic expansion for low densities. \endtheo \bigskip -Thus the Simple Equation reproduces the same prediction\-~(\ref{lhy_h}) as Bogolyubov theory. +\indent +This statement is proven in\-~\cite{CJL21}. +Here, let us present a different proof: the proof in\-~\cite{CJL21} is a bit more general, but uses a more sophisticated discussion. +Instead, here we will present a more elementary proof that has slightly different requirements on $v$. +\bigskip + +\theo{Theorem} + For the Simple Equation, in $d=3$, suppose $v\in L_1(\mathbb R^3)\cap L_2(\mathbb R^3)$, $v\geqslant 0$, and $\hat v$ is H\"older continuous: there exist $C,\alpha>0$ such that, for $k,q\in\mathbb R^3$, $|\hat v(k+q)-\hat v(k)|\leqslant C|q|^\alpha$. + As $\rho\to0$, + \nopagebreakaftereq + \begin{equation} + 1-\eta\sim\frac{8\sqrt{\rho a^3}}{3\sqrt\pi} + . + \end{equation} +\endtheo +\restorepagebreakaftereq +\bigskip + +\indent\underline{Proof}: + We write\-~(\ref{simpleq_eta}) in Fourier space: + \begin{equation} + \rho\hat u_\epsilon(k)=\left(\frac{k^2+2\epsilon}{4e_\epsilon}+1\right) + \left(1-\sqrt{1-\frac{\frac\rho{2e_\epsilon}\hat S_\epsilon(k)}{(\frac{k^2+2\epsilon}{4e_\epsilon}+1)^2}}\right) + ,\quad + \hat S_\epsilon(k):=\int dx\ e^{ikx}(1-u_\epsilon(x))v(x) + . + \label{fmu} + \end{equation} + Thus + \begin{equation} + \rho\partial_\epsilon\hat u_\epsilon(k)|_{\epsilon=0} + = + \frac1{2e_0}\left( + 1-\frac{k^2}{2e_0}\eta + - + \frac{(\frac{k^2}{4e_0}+1)(1-\frac{k^2}{2e_0}\eta)+\frac\rho{2e_0}\eta\hat S_0(k)-\frac\rho2\partial_\epsilon\hat S_\epsilon(k)|_{\epsilon=0}}{\sqrt{(\frac{k^2}{4e_0}+1)^2-\frac \rho{2e_0}\hat S_0(k)}} + \right) + . + \label{du} + \end{equation} + Furthermore, + \begin{equation} + \eta=-\frac12\int\frac{dk}{8\pi^3}\ \hat v(k)\rho\partial_\epsilon\hat u_\epsilon(k)|_{\epsilon=0} + \end{equation} + which we split into three terms: + \begin{equation} + \eta=H_1+H_2+H_3 + \label{etasplit} + \end{equation} + with + \begin{equation} + H_1:=\frac1{32e_0\pi^3}\int dk\ \hat v(k) + \left( + \frac{\frac{k^2}{4e_0}+1}{\sqrt{(\frac{k^2}{4e_0}+1)^2-\frac \rho{2e_0}\hat S_0(k)}}-1 + \right) + \end{equation} + \begin{equation} + H_2:=-\eta\frac1{64e_0^2\pi^3}\int dk\ \hat v(k) + k^2\left( + \frac{\frac{k^2}{4e_0}+1-\frac{\rho}{k^2}\hat S_0(k)}{\sqrt{(\frac{k^2}{4e_0}+1)^2-\frac \rho{2e_0}\hat S_0(k)}} + -1 + \right) + \end{equation} + \begin{equation} + H_3:=-\frac1{32e_0\pi^3}\int dk\ \hat v(k) + \frac{\frac\rho2\partial_\epsilon\hat S_\epsilon(k)|_{\epsilon=0}}{\sqrt{(\frac{k^2}{4e_0}+1)^2-\frac \rho{2e_0}\hat S_0(k)}} + . + \end{equation} + \bigskip + + \point We first compute $H_1$. + We rescale the integral by $2\sqrt{e_0}$: + \begin{equation} + H_1=\frac{\sqrt{e_0}}{4\pi^3}\int dk\ \hat v(2\sqrt{e_0}k) + \left( + \frac{k^2+1}{\sqrt{(k^2+1)^2-\frac \rho{2e_0}\hat S_0(2\sqrt{e_0}k)}}-1 + \right) + . + \end{equation} + Since $S_0(x)\geqslant 0$, $|\hat S_0(k)|\leqslant|\hat S_0(0)|=\frac{2e_0}\rho$, so + \begin{equation} + \left|\frac{k^2+1}{\sqrt{(k^2+1)^2-\frac \rho{2e_0}\hat S_0(2\sqrt{e_0}k)}}-1\right| + \leqslant + \frac{k^2+1}{\sqrt{(k^2+1)^2-1}}-1 + \label{ineqH1} + \end{equation} + which is integrable, and since $v(x)\geqslant 0$, $|\hat v(k)|\leqslant|\hat v(0)|$, so, by dominated convergence, + \begin{equation} + H_1\sim\frac{\sqrt{e_0}}{4\pi^3}\hat v(0)\int dk + \left(\frac{k^2+1}{\sqrt{(k^2+1)^2-1}}-1\right) + = + \frac{\sqrt{2e_0}}{3\pi^2}\hat v(0) + . + \label{H1} + \end{equation} + ($\hat v$ is continuous since $v\in L_1(\mathbb R^3)$, see e.g. \cite[Section\-~5.1]{LL01}.) + \bigskip + + \point We now turn to $H_2$. + We rescale the integral by $2\sqrt{e_0}$: + \begin{equation} + H_2= + -\eta\frac{\sqrt{e_0}}{2\pi^3}\int dk\ \hat v(2\sqrt{e_0}k) + k^2\left( + \frac{k^2+1-\frac\rho{4e_0k^2}\hat S_0(2\sqrt e_0k)}{\sqrt{(k^2+1)^2-\frac \rho{2e_0}\hat S_0(2\sqrt{e_0}k)}}-1 + \right) + . + \end{equation} + Furthermore, consider the function + \begin{equation} + \mathfrak G_k(\sigma):= + \frac{k^2+1-\frac\sigma{2k^2}}{\sqrt{(k^2+1)^2-\sigma}}-1 + \end{equation} + its critical points $\partial_\sigma\mathfrak G_k(\sigma)=0$ satisfy + \begin{equation} + \partial_\sigma\mathfrak G_k(\sigma)=\frac{k^2+1-\frac\sigma{2k^2}}{2((k^2+1)^2-\sigma)^{\frac32}}-\frac1{2k^2\sqrt{(k^2+1)^2-\sigma}}=0 + \end{equation} + that is + \begin{equation} + \sigma=2(1+k^2)>1 + . + \end{equation} + In addition, + \begin{equation} + \partial_\sigma\mathfrak G_k(0) + = + -\frac1{2k^2(k^2+1)^2}<0 + \end{equation} + so $\mathfrak G_k(\sigma)$ is decreasing and negative for $|\sigma|\leqslant 1$. + Therefore, + \begin{equation} + |\mathfrak G_k(\sigma)|\leqslant|\mathfrak G_k(1)| + \end{equation} + and, since $\hat S_0\leqslant\frac{2e_0}\rho$, + \begin{equation} + k^2\left| + \frac{k^2+1-\frac\rho{4e_0k^2}\hat S_0(2\sqrt e_0k)}{\sqrt{(k^2+1)^2-\frac \rho{2e_0}\hat S_0(2\sqrt{e_0}k)}}-1 + \right| + \leqslant + k^2\left| + \frac{k^2+1-\frac1{2k^2}}{\sqrt{(k^2+1)^2-1}}-1 + \right| + \label{ineqH2} + \end{equation} + which is integrable. + Therefore, by dominated convergence, + \begin{equation} + H_2\sim + -\eta\frac{\sqrt{e_0}}{2\pi^3}\hat v(0)\int dk\ + k^2\left( + \frac{k^2+1-\frac1{2k^2}}{\sqrt{(k^2+1)^2-1}}-1 + \right) + =O(\eta\sqrt{e_0}) + . + \label{H2} + \end{equation} + \bigskip + + \point We now turn to $H_3$. + We expand + \begin{equation} + \frac1{\sqrt{(\frac{k^2}{4e_0}+1)^2-\frac \rho{2e_0}\hat S_0(k)}} + = + \frac1{\frac{k^2}{4e_0}+1}(1+O(e_0^2k^{-4})) + = + \frac{4e_0}{k^2}(1+O(e_0k^{-2})) + . + \end{equation} + We extract the non-integrable term by defining + \begin{equation} + H_3=h_3- + \frac1{8\pi^3}\int dk\ \hat v(k) + \frac{\frac\rho{2}\partial_\epsilon\hat S_\epsilon(k)|_{\epsilon=0}}{k^2} + \label{H3split} + \end{equation} + with + \begin{equation} + h_3:=\frac1{32e_0\pi^3}\int dk\ \hat v(k) + \frac\rho2\partial_\epsilon\hat S_\epsilon(k)|_{\epsilon=0}\mathfrak R_3(k) + ,\quad + \mathfrak R_3(k):=\frac{4e_0}{k^2}-\frac1{\sqrt{(\frac{k^2}{4e_0}+1)^2-\frac \rho{2e_0}\hat S_0(k)}} + . + \end{equation} + \bigskip + + \subpoint To compute $h_3$, we rescale the integral by $2\sqrt{e_0}$: + \begin{equation} + h_3=\frac{\sqrt{e_0}}{4\pi^3}\int dk\ \hat v(2\sqrt{e_0}k) + \frac\rho2\partial_\epsilon\hat S_\epsilon(2\sqrt{e_0}k)|_{\epsilon=0}\mathfrak R_3(2\sqrt{e_0}k) + \end{equation} + and + \begin{equation} + 0\leqslant \mathfrak R_3(2\sqrt{e_0}k)= + \frac1{k^2}-\frac1{\sqrt{(k^2+1)^2-\frac \rho{2e_0}\hat S_0(2\sqrt{e_0}k)}} + \leqslant + \frac1{k^2}- + \frac1{\sqrt{(k^2+1)^2+1}} + \label{ineqH3} + \end{equation} + which is integrable, so, using the fact that + \begin{equation} + \eta=\frac\rho2\partial_\epsilon\hat S_\epsilon(0)|_{\epsilon=0} + \end{equation} + by dominated convergence, + \begin{equation} + h_3\sim + \frac{\eta\sqrt{e_0}}{4\pi^3}\hat v(0)\int dk\ \left(\frac1{k^2}-\frac1{\sqrt{(k^2+1)^2-1}}\right) + =O(\eta\sqrt{e_0}) + . + \label{h3} + \end{equation} + \bigskip + + \subpoint Furthermore, + \begin{equation} + \frac\rho2\partial_\epsilon\hat S_\epsilon(q)|_{\epsilon=0}=-\frac\rho2\int\frac{dk}{8\pi^3}\ \hat v(k-q)\partial_\epsilon\hat u_\epsilon(k)|_{\epsilon=0} + \end{equation} + so, by\-~(\ref{du}), + \begin{equation} + \frac\rho2\partial_\epsilon\hat S_\epsilon(q)|_{\epsilon=0}=F_1(q)+F_2(q)+F_3(q) + \label{dSsplit} + \end{equation} + with + \begin{equation} + F_1(q):=\frac1{32e_0\pi^3}\int dk\ \hat v(k-q) + \left( + \frac{\frac{k^2}{4e_0}+1}{\sqrt{(\frac{k^2}{4e_0}+1)^2-\frac \rho{2e_0}\hat S_0(k)}}-1 + \right) + \label{F1} + \end{equation} + \begin{equation} + F_2(q):=-\eta\frac1{64e_0^2\pi^3}\int dk\ \hat v(k-q) + k^2\left( + \frac{\frac{k^2}{4e_0}+1-\frac{\rho}{k^2}\hat S_0(k)}{\sqrt{(\frac{k^2}{4e_0}+1)^2-\frac \rho{2e_0}\hat S_0(k)}} + -1 + \right) + \label{F2} + \end{equation} + \begin{equation} + F_3(q):=-\frac1{32e_0\pi^3}\int dk\ \hat v(k-q) + \frac{\frac\rho2\partial_\epsilon\hat S_\epsilon(k)|_{\epsilon=0}}{\sqrt{(\frac{k^2}{4e_0}+1)^2-\frac \rho{2e_0}\hat S_0(k)}} + . + \label{F3} + \end{equation} + Now, let + \begin{equation} + r_1(q):=F_1(q)-\frac{\sqrt{2e_0}}{3\pi^2}\hat v(q) + ,\quad + r_2(q):=F_2(q) + \end{equation} + \begin{equation} + r_3(q):=F_3(q) + +\frac1{8\pi^3}\int dk\ \hat v(k-q)\frac{\frac\rho2\partial_\epsilon\hat S_\epsilon(k)|_{\epsilon=0}}{k^2} + . + \end{equation} + Furthermore, by\-~(\ref{ineqH2}), and using the fact that $|\hat v(k)|\leqslant\hat v(0)$, + \begin{equation} + |r_2(q)| + \leqslant + |\eta|\frac{\sqrt{e_0}}{2\pi^3}\hat v(0)\int dk\ + k^2\left| + \frac{k^2+1-\frac1{2k^2}}{\sqrt{(k^2+1)^2-1}}-1 + \right| + =O(\eta\sqrt{e_0}) + \label{r2} + \end{equation} + uniformly in $q$. + Similarly, by\-~(\ref{ineqH3}), + \begin{equation} + |r_3(q)|\leqslant + \frac{\sqrt{e_0}}{4\pi^3}\hat v(0)\int dk\ \frac\rho2\partial_\epsilon\hat S_\epsilon(2\sqrt{e_0}k)|_{\epsilon=0}\left|\frac1{\sqrt{(k^2+1)^2-1}}-\frac1{k^2}\right| + =O(\eta\sqrt{e_0}) + \label{r3} + \end{equation} + uniformly in $q$. + Bounding $r_1$ is a bit more of a challenge. + By\-~(\ref{H1}), + \begin{equation} + \begin{largearray} + r_1(q)= + \frac{\sqrt{e_0}}{4\pi^3}\int dk\ (\hat v(q-2\sqrt{e_0}k)-\hat v(q)) + \left( + \frac{k^2+1}{\sqrt{(k^2+1)^2-\frac \rho{2e_0}\hat S_0(2\sqrt{e_0}k)}}-1 + \right) + \\[0.5cm]\hfill + + + \hat v(q)\frac{\sqrt{e_0}}{4\pi^3}\int dk\ + \left( + \frac{k^2+1}{\sqrt{(k^2+1)^2-\frac \rho{2e_0}\hat S_0(2\sqrt{e_0}k)}} + -\frac{k^2+1}{\sqrt{(k^2+1)^2-1}} + \right) + . + \end{largearray} + \end{equation} + Furthermore, by\-~(\ref{ineqH1}), and using the H\"older continuity of $\hat v$, + \begin{equation} + \begin{largearray} + \frac{\sqrt{e_0}}{4\pi^3}\int dk\ |\hat v(q-2\sqrt{e_0}k)-\hat v(q)| + \left| + \frac{k^2+1}{\sqrt{(k^2+1)^2-\frac \rho{2e_0}\hat S_0(2\sqrt{e_0}k)}}-1 + \right| + \\[0.5cm]\hfill + \leqslant + C(2\sqrt{e_0})^{\alpha}\frac{\sqrt{e_0}}{4\pi^3}\int dk\ + k^\alpha\left| + \frac{k^2+1}{\sqrt{(k^2+1)^2-1}}-1 + \right| + =o(\sqrt{e_0}) + . + \end{largearray} + \end{equation} + uniformly in $q$. + Furthermore, + \begin{equation} + \left| + \frac{k^2+1}{\sqrt{(k^2+1)^2-\frac \rho{2e_0}\hat S_0(2\sqrt{e_0}k)}} + -\frac{k^2+1}{\sqrt{(k^2+1)^2-1}} + \right| + \leqslant + \frac{k^2+1}{((k^2+1)^2-1)^{\frac32}}\frac{1-\frac\rho{2e_0}\hat S_0(2\sqrt{e_0}k)}2 + . + \end{equation} + Now, + \begin{equation} + \hat S_0(k)=\hat v(k)-\hat u_0\ast\hat v(k) + \end{equation} + so + \begin{equation} + |\hat S_0(k+\epsilon)-\hat S_0(k)| + \leqslant + |\hat v(k+\epsilon)-\hat v(k)| + +\int\frac{dq}{8\pi^3}|\hat v(k+\epsilon-q)-\hat v(k-q)||\hat u_0(q)| + \end{equation} + and + \begin{equation} + |\hat S_0(k+\epsilon)-\hat S_0(k)| + \leqslant + C\epsilon^\alpha\left(1+\int\frac{dq}{8\pi^3}|\hat u_0(q)|\right) + \end{equation} + and, since $u_0(x)\leqslant 1$, $\|\hat u_0\|_1\leqslant 8\pi^3$, so $\hat S_0$ is H\"older continuous. + Therefore + \begin{equation} + \begin{largearray} + \frac{\sqrt{e_0}}{4\pi^3}|\hat v(q)|\int dk\ + \left| + \frac{k^2+1}{\sqrt{(k^2+1)^2-\frac \rho{2e_0}\hat S_0(2\sqrt{e_0}k)}} + -\frac{k^2+1}{\sqrt{(k^2+1)^2-1}} + \right| + \\[0.5cm]\hfill + \leqslant + C\sqrt{2e_0}^{\alpha}\frac{\rho}{2e_0} + \frac{\sqrt{e_0}}{4\pi^3}|\hat v(q)|\int dk\ + k^\alpha\frac{k^2+1}{((k^2+1)^2-1)^{\frac32}} + =o(\sqrt{e_0}) + \end{largearray} + \end{equation} + uniformly in $q$. + Thus, + \begin{equation} + r_1(q)=o(\sqrt{e_0}) + \label{r1} + \end{equation} + uniformly in $q$. + \bigskip + + \subpoint By inserting\-~(\ref{r1}), (\ref{r2}) and\-~(\ref{r3}) into\-~(\ref{dSsplit}), we find + \begin{equation} + \frac\rho2\partial_\epsilon\hat S_\epsilon(q)|_{\epsilon=0} + + + \frac1{8\pi^3}\int dk\ \hat v(k-q)\frac{\frac\rho2\partial_\epsilon\hat S_\epsilon(k)|_{\epsilon=0}}{k^2} + = + \frac{\sqrt{2e_0}}{3\pi^2}\hat v(q) + +o(\sqrt{e_0}) + +O(\eta\sqrt{e_0}) + \end{equation} + where $o(\sqrt{e_0})$ and $O(\eta\sqrt{e_0})$ are uniform in $q$. + In other words, if we define + \begin{equation} + \hat f(k):= + \frac{3\pi^2}{\sqrt{2e_0}}\frac{\frac\rho2\partial_\epsilon\hat S_\epsilon(k)|_{\epsilon=0}}{k^2} + \end{equation} + we have + \begin{equation} + (k^2+\hat v\ast)\hat f(k)=\hat v(k)+o(1)+O(\eta) + . + \end{equation} + Now, denoting the solution of the scattering equation\-~(\ref{scateq_varphi}) by $\varphi$: + \begin{equation} + (-\Delta+v)\varphi(x)=v(x) + \end{equation} + and its Fourier transform by $\hat\varphi$, we have + \begin{equation} + -\frac1{8\pi^3}\int dk\ \hat v(k) + \frac{\frac\rho{2}\partial_\epsilon\hat S_\epsilon(k)|_{\epsilon=0}}{k^2} + = + -\frac{\sqrt{2e_0}}{3\pi^2}\int \frac{dk}{8\pi^3}\ \hat v(k)\hat f(k) + = + -\frac{\sqrt{2e_0}}{3\pi^2}\int \frac{dk}{8\pi^3}\ (k^2+\hat v\ast)\hat\varphi(k)\hat f(k) + \end{equation} + and since $(k^2+\hat v\ast)$ is symmetric, + \begin{equation} + -\frac1{8\pi^3}\int dk\ \hat v(k) + \frac{\frac\rho{2}\partial_\epsilon\hat S_\epsilon(k)|_{\epsilon=0}}{k^2} + = + -\frac{\sqrt{2e_0}}{3\pi^2}\int \frac{dk}{8\pi^3}\ \hat v(k)\hat\varphi(k) + -\int\frac{dk}{8\pi^3}\ \hat\varphi(k)(o(\sqrt{e_0})+O(\sqrt{e_0}\eta)) + . + \end{equation} + and so + \begin{equation} + -\frac1{8\pi^3}\int dk\ \hat v(k) + \frac{\frac\rho{2}\partial_\epsilon\hat S_\epsilon(k)|_{\epsilon=0}}{k^2} + = + -\frac{\sqrt{2e_0}}{3\pi^2}\int \frac{dk}{8\pi^3}\ \hat v(k)\hat\varphi(k) + +(o(\sqrt{e_0})+O(\sqrt{e_0}\eta)) + . + \end{equation} + Finally, by Lemma\-~\ref{lemma:scattering} + \begin{equation} + \int \frac{dk}{8\pi^3}\ \hat v(k)\hat\varphi(k) + = + \int dx\ v(k)\varphi(k) + = + \hat v(0)-4\pi a + . + \end{equation} + Therefore, by\-~(\ref{H3split}) and\-~(\ref{h3}), + \begin{equation} + H_3= + \frac{\sqrt{2e_0}}{3\pi^2}(4\pi a-\hat v(0)) + +o(\sqrt{e_0})+O(\sqrt{e_0}\eta) + . + \label{H3} + \end{equation} + Inserting\-~(\ref{H1}), (\ref{H2}) and\-~(\ref{H3}) into\-~(\ref{etasplit}), we find + \begin{equation} + \eta\sim\frac{4\sqrt{2e_0}}{3\pi}a + . + \end{equation} + We conclude the proof using Theorem\-~\ref{theo:energy}. +\qed + +\subsection{Other observables} +\indent +As we have seen, the Simple equation makes predictions for the ground state energy that agree with that of the many-body interacting Boson system for high and low densities, and its prediction for the condensate fraction agrees with Bogolyubov theory at low density. +We have studied the predictions of the Simple equation for other observables as well. +We will state these in this section, but will not discuss the proofs. +Instead, we will refer the readers to the appropriate publications for details. +\bigskip + +\subsubsection{Two-point correlation function} +\indent +The two-point correlation function is the joint probability (in the usual quantum mechanical probability distribution $|\psi|^2$, not the probability distribution $\psi/\int\psi$ that we considered in Chapter\-~\ref{sec:simplified_def}) of finding a particle at $y$ and another at $y'$: +\begin{equation} + C_2^{(0)}(y-y'):= + \lim_{\displaystyle\mathop{\scriptstyle N,V\to\infty}_{\frac NV=\rho}} + \sum_{i\neq j}\left<\psi_0\right|\delta(x_i-y)\delta(x_j-y')\left|\psi_0\right> + . + \label{C2} +\end{equation} +Lee, Huang and Yang made the following prediction, using Bogolyubov theory. +\bigskip + +\theoname{Conjecture}{{\rm\cite[(48)]{LHY57}}}\label{conjecture:lhy_C2} + For any potential $v\in L_1(\mathbb R^3)$ with $v\geqslant 0$ (actually, the potential may include a hard-core component), as $\sqrt{\rho a}|x|\to \infty$, + \begin{equation} + \frac1{\rho^2}C_2^{(0)}(x)-1\sim\frac{16\rho a^3}{\pi^3(\sqrt{\rho a}|x|)^4} + \end{equation} + where $a$ is the scattering length of the potential. +\endtheo \bigskip \indent -The proof of this theorem is a bit more involved than the computation of the energy, though it is similar in spirit. -We refer interested readers to\-~\cite{CJL21} for details. +As was the case for the condensate fraction, see Section\-~\ref{sec:condensate_fraction}, we must define a natural prediction of the Simple equation for the two-point correlation function. +As we mentioned above, the Simple equation does not predict the ground state wavefunction $\psi_0$, so we must relate the correlation function to the ground state energy. +To do so, we simplify\-~(\ref{C2}) by taking advantage of the translation invariance: if $z=y-y'$, then +\begin{equation} + \sum_{i\neq j}\left<\psi_0\right|\delta(x_i-y)\delta(x_j-y')\left|\psi_0\right> + = + \sum_{i\neq j}\left<\psi_0\right|\delta(x_i-z-y')\delta(x_j-y')\left|\psi_0\right> +\end{equation} +but, by translation invariance, this quantity is independent of $y'$, so we can take an average over $y'$: +\begin{equation} + \begin{largearray} + \sum_{i\neq j}\left<\psi_0\right|\delta(x_i-y)\delta(x_j-y')\left|\psi_0\right> + = + \frac1V\int dy' + \sum_{i\neq j}\left<\psi_0\right|\delta(x_i-z-y')\delta(x_j-y')\left|\psi_0\right> + =\\\hfill= + \frac1V\sum_{i\neq j}\left<\psi_0\right|\delta(x_i-x_j-z)\left|\psi_0\right> + \end{largearray} +\end{equation} +that is, +\begin{equation} + \sum_{i\neq j}\left<\psi_0\right|\delta(x_i-y)\delta(x_j-y')\left|\psi_0\right> + = + \frac2V\sum_{i +\end{equation} +which is +\begin{equation} + \sum_{i\neq j}\left<\psi_0\right|\delta(x_i-y)\delta(x_j-y')\left|\psi_0\right> + = + \frac2V\frac{\delta}{\delta v(z)}\left<\psi_0\right|H\left|\psi_0\right> +\end{equation} +and so +\begin{equation} + C_2^{(0)}(z)=2\rho\frac{\delta e_0}{\delta v(z)} + . +\end{equation} +\bigskip + +\indent +Therefore, the prediction of the Simple equation for the two-point correlation function is defined to be +\begin{equation} + C_2(z):=2\rho\frac{\delta e}{\delta v(z)} + . +\end{equation} +This quantity can be proved to decay as $|x|^{-4}$ for any value of the density, which agrees with the Bogolyubov prediction in Conjecture\-~\ref{conjecture:lhy_C2} (though the result for the Simple equation goes beyond, as it is not restricted to small densities). +\bigskip + +\theoname{Theorem}{{\rm\cite[Theorem 4.5]{Ja22}}}\label{theo:C2} + Under the assumptions of Theorem\-~\ref{theo:existence}, if $(1+|x|^6)v(x)\in L_1(\mathbb R^d)$, then + \begin{equation} + \lim_{|x|\to\infty}|x|^4\left|\frac{C_2}{\rho^2}-1-r(x)\right|<\infty + \end{equation} + where $|x|^4r\in L_2(\mathbb R^3)\cap L_\infty(\mathbb R^3)$. +\endtheo +\bigskip + +\indent +Strictly speaking, this does not necessarily mean that $C_2\sim|x|^{-4}$: the function $|x|^4r$ is not guaranteed to decay. +However it is square integrable and bounded, so, at worst, there could be small intervals where $|x|^4r$ is not small (though also not large), and these intervals would get more and more spaced out as $|x|\to\infty$. +This is not far from saying that $r$ decays faster than $|x|^{-4}$. +\bigskip + +\indent +The proof uses some tools that we have not introduced here. +Namely, it relies on the analysis of the operator +\begin{equation} + \mathfrak K_e:=(-\Delta+v+4e(1-\rho u\ast))^{-1} + . +\end{equation} +This operator is studied in depth in\-~\cite{CJL21}. +The full proof of Theorem\-~\ref{theo:C2} can be found in\-~\cite[Appendix B]{Ja22}. + +\subsubsection{Momentum distribution} +\indent +The condensate fraction is the proportion of particles in the condensate state, which is the constant state. +The constant state can also be seen as the zero-momentum state: $e^{ikx}|_{k=0}$. +A natural extension is to compute the probability of finding a particle in a state $e^{ikx}$ with momentum $k$. +This quantity is called the {\it momentum distribution}, and is defined as +\begin{equation} + \mathcal M_0(k):= + \lim_{\displaystyle\mathop{\scriptstyle N,V\to\infty}_{\frac NV=\rho}} + \frac1N\sum_{i=1}^N\left<\psi_0\right|K_i(k)\left|\psi_0\right> + \label{momentum_distribution_def} +\end{equation} +where $K_i(k)$ is the projector on the subspace where particle $i$ has momentum $k$: +\begin{equation} + K_i(k)\psi(x_1,\cdots,x_N):=e^{ikx_i}\int dy_i\ e^{-iky_i}\psi(x_1,\cdots,x_{i-1},y_i,x_{i+1},\cdots,x_N) + . +\end{equation} +The prediction from Bogolyubov theory can be recovered from\-~\cite[Appendix\-~A]{LSe05}: +\bigskip + +\theo{Conjecture}\label{conjecture:Nk} + For any potential $v\in L_1(\mathbb R^3)$ with $v\geqslant 0$ (actually, the potential may include a hard-core component), as $\rho\to0$ and + \begin{equation} + \frac{|k|}{\sqrt{8m\pi\rho a}}=:\kappa + \label{kappak_conj} + \end{equation} + is fixed, + \begin{equation} + \mathcal M_0(k)\sim\frac1{2\rho}\left(\frac{\kappa^2+1}{\sqrt{(\kappa^2+1)^2-1}}-1\right) + \label{Nk_conj} + \end{equation} + where $a$ is the scattering length of the potential. +\endtheo +\bigskip + +See Appendix\-~\ref{app:bog_Nk} for a derivation of this expression from\-~\cite[Appendix\-~A]{LSe05}. +\bigskip + +\indent +To obtain a prediction for the momentum distribution using the Simple equation, we proceed in a similar way as for the condensate fraction: we add a term to the Hamiltonian +\begin{equation} + \tilde H_N(\epsilon):=H_N+\epsilon\sum_{i=1}^NK_i(k) +\end{equation} +after which +\begin{equation} + \mathcal M_0(k)= + \lim_{\displaystyle\mathop{\scriptstyle N,V\to\infty}_{\frac NV=\rho}} + \frac1N\partial_\epsilon \tilde E_0|_{\epsilon=0} + . +\end{equation} +We then repeat the arguments from Sections\-~\ref{sec:simplified_construction}-\ref{sec:equations} and Appendix\-~\ref{app:proof_factorization} to the modified Hamiltonian. +However, there is one added difficulty: the modified Hamiltonian no longer necessarily has a unique ground state. +In particular, it is not guaranteed that its ground states are translation invariant. +However, translation invariance is used repeatedly in the derivation of Lieb's Simplified approach. +In order to carry this out carefully, this derivation needs to be adapted to cases without translation invariance. +This was done in\-~\cite{Ja24}. +The modified Simple equation is\-~\cite[(60)]{Ja24}: +\begin{equation} + -\Delta u_\epsilon=(1-u_\epsilon)v-4e_\epsilon u_\epsilon+2\rho e_\epsilon u_\epsilon\ast u_\epsilon+\epsilon F(x) + ,\quad + e_\epsilon:=\frac\rho2\int dx\ (1-u_\epsilon(x))v(x) +\end{equation} +with +\begin{equation} + F(x):=-2\hat u_\epsilon(k)\cos(kx) +\end{equation} +in terms of which +\begin{equation} + \mathcal M(k):=\partial_\epsilon e_\epsilon|_{\epsilon=0} + . +\end{equation} +\bigskip + +\theoname{Theorem}{{\rm\cite[Theorem 3]{Ja24}}}\label{theo:Nk} + Under the assumptions of Theorem\-~\ref{theo:existence}, for $d=3$, if + \begin{equation} + \kappa:=\frac{|k|}{2\sqrt{e}} + \end{equation} + then, for all $\kappa\in\mathbb R^3$, + \begin{equation} + \lim_{\rho\to0}\rho\mathcal M(2\sqrt{e}\kappa)=\frac12\left(\frac{\kappa^2+1}{\sqrt{(\kappa^2+1)^2-1}}-1\right) + . + \label{Nk} + \end{equation} +\endtheo +\bigskip + +The proof of this Theorem can be found in\-~\cite[Section\-~4.1]{Ja24}. +\bigskip + +\indent +The scaling $|k|\sim\sqrt{8m\pi\rho a}$ in Conjecture\-~\ref{conjecture:Nk} comes from Theorem\-~\ref{theo:Nk} (using the leading order in\-~(\ref{lhy})), which suggests that the asymptotic behavior in\-~(\ref{Nk_conj}) holds for values of $\sqrt{\rho a}\lesssim |k|\ll 1$. +Now, if we were to take the limit $\kappa\to\infty$ {\it after} the limit $\rho\to0$ in\-~(\ref{Nk}), then we find +\begin{equation} + \lim_{\rho\to0}\rho\mathcal M(2\sqrt e\kappa)\sim_{\kappa\to\infty}\frac1{4\kappa^4}\sim\frac{4e^2}{|k|^4}\sim\frac{16\pi^2m^2\rho^2 a^2}{|k|^4} + . +\end{equation} +The asymptotics $\rho\mathcal M_0(k)\sim\frac C{|k|^4}$ is called the {\it universal Tan relation}, which was first derived for Fermions\-~\cite{Ta08,Ta08b,Ta08c}, and extended to Bosonic systems\-~\cite{CAL09}. +In fact, the value $C=16\pi^2m^2\rho^2a^2$ was predicted by\-~\cite[6.2.1.2]{NE17} from\-~\cite{CAL09}. +Thus the Simple equation reproduces the universal Tan relation, but it also shows that the order of the limits matters when considering low density and large $|k|$. +Indeed, it predicts that the Tan relation holds if $\rho\to0$ with $\frac{|k|}{\sqrt{8m\pi\rho a}}$ fixed (which means that $|k|$ goes to $0$ with $\rho$), and {\it then}, $\kappa$ is taken to infinity. +In other words, the Simple equation predicts that the Tan relation holds in the regime $\sqrt{\rho a}\ll|k|\ll 1$. +Now, this is only relevant for low-densities: if $\sqrt{\rho a}$ is not small, then the range $\sqrt{\rho a}\ll|k|\ll 1$ is empty. +In fact, it was verified numerically, using the Medium and Big equations (which are more accurate for intermediate densities), that the Tan relation does {\it not} hold in the intermediate density regime\-~\cite[Fig.\-~6]{CHe21}. + +\subsubsection{Decay of the solution $u$} +\indent +A useful lemma that was used to compute the decay of two-point correlation function, as well as several other properties of the solution of the Simple equation is an estimate of the decay rate of the solution $u$. +\bigskip + +\theoname{Lemma}{{\rm\cite[Theorem 1.2]{CJL21}}} + In $d=3$, under the assumptions of Theorem\-~\ref{theo:existence}, if $(1+|x|^4)v(x)\in L_1(\mathbb R^3)\cap L_2(\mathbb R^3)$, then + \begin{equation} + \rho u(x)=\frac{\sqrt{2+\beta}}{2\pi^2\sqrt e}\frac1{|x|^4}+R(x) + \end{equation} + where + \begin{equation} + \beta=\rho\int dx\ |x|^2v(x)(1-u(x))dx\leqslant\rho\||x|^2v\|_1 + \end{equation} + and $|x|^4R(x)\in L_2(\mathbb R^3)\cap L_\infty(\mathbb R^3)$, uniformly in $e$ on all compact sets. + In addition, $\forall\rho_0>0$, there exists a constant $C$ that only depends on $\rho$ such that, for all $\rho<\rho_0$ and all $x\in\mathbb R^3$, + \nopagebreakaftereq + \begin{equation} + u(x)\leqslant\min\left\{1,\frac C{\rho\sqrt e|x|^4}\right\} + . + \end{equation} +\endtheo +\restorepagebreakaftereq +\bigskip + +The proof of this Theorem can be found in\-~\cite[Section\-~2]{CJL21}. +\bigskip + +\indent +The decay of $u$ translates to the decay of the two-point correlation function in Theorem\-~\ref{theo:C2}, see\-~\cite{Ja22}, and is used to prove various results in\-~\cite{CJL21}, as it is used to prove bounds on the operator $\mathfrak K_e$, see\-~\cite[Section\-~1.2]{CJL21}. + +\section{Numerical computation of the solution to the Big and Medium equations}\label{sec:numerics} +\indent +We have seen that the predictions of the Simple equation can be studied analytically. +However, as one can see from Figures\-~\ref{fig:energy}-\ref{fig:condensate}, the Simple equation is not so accurate in the intermediate density regime. +The predictions of the Big and Medium equations are much more accurate. +However, no analytical results about those equations have been proved so far: even the existence of a solution remains an outstanding open problem, see Chapter\-~\ref{sec:open}. +On the other hand, these equations are PDEs in three dimensions, and solutions can be computed numerically with modest hardware. +\bigskip + +\indent +This numerical computation has been implemented in a software package called {\tt simplesolv}\-~\cite{ss} that is available for use under the free Apache License. +The program is written in the Julia language\-~\cite{Julia}. +Documentation is bundled with the software package that explains the details of the numerical algorithms used, and their accuracy. +\bigskip + +\indent +The numerical computation is carried out in Fourier space, as the equations of Lieb's Simplified approach have fewer convolutions in Fourier space, and thus fewer numerical integrals need to be evaluated. +The equations are then written in the form +\begin{equation} + \hat u(k)=F(\hat u)(k) + \label{newton_setup} +\end{equation} +where $F$ is a functional that is specific to each equation of the approach. +Note that we are searching for spherically symmetric solutions of the equations, because $v$ is spherically symmetric, so the unknown function $\hat u$ is a function of $|k|$ alone. +Next, we discretize the space of functions $\hat u(|k|)$ by truncating its Chebyshev polynomial expansion. +In doing so, we represent the function $\hat u$ as a finite vector, from which we can recover an approximation for $\hat u$ in terms of Chebyshev polynomials. +The approximation is exponentially accurate if $\hat u$ is analytic (see the documentation for\-~\cite{ss}). +(Note that $\hat u(k)$ is not analytic, as $u(x)\sim|x|^{-4}$, but $|k|\mapsto \hat u(|k|)$ presumably is.) +Some terms in $F$ are integrals of $\hat u$. +To evaluate them, we approximate the integrals using Gauss quadratures. +This approximation is also exponentially accurate if $\hat u$ is analytic\-~\cite{ss}. +Finally, to solve\-~(\ref{newton_setup}) (which, having discretized space as we did, is now a vector valued equation), we use the Newton algorithm, which converges super-exponentially fast. +\bigskip + +\indent +In this way, we can compute numerical values for various observables for any of the equations of Lieb's Simplified approach. +Note that it is significantly faster to run the computation for the Medium equation than the Big equation (see the documentation of\-~\cite{ss} for an explanation), so it provides a good middle ground between computational complexity and accuracy. +\bigskip + +\indent +The plots for the energy and condensate fraction were already discussed above, see Figures\-~\ref{fig:energy}-\ref{fig:condensate}. +In both cases we see that the simple equation is accurate at low and high density, but rather far off the mark in the intermediate density regime. +On the other hand, the Big equation is remarkably accurate (especially so for the energy) at {\it all} densities. +The Medium equation lies in the middle of the Simple and Big. +(Note that we have found that larger potentials lead to worse accuracy\-~\cite{CHe21}.) +The plot of the condensate fraction further reveals an interesting fact: the condensate fraction reaches a minimum at intermediate densities. +\bigskip + +\indent +A computation of the two-point correlation function is shown in Figure\-~\ref{fig:2pt}. +As was the case for the energy and condensate fraction, the prediction of the Simple equation is only accurate at small densities (and values of $|x|$ that are not too small). +On the other hand, the Big equation reproduces the results of the Quantum Monte Carlo computation rather accurately at all densities. +In particular, we see that the correlation function exhibits a local maximum above $\rho^2$ at intermediate densities. +This is a rather interesting fact: such a local maximum implies the existence of a preferred length scale, at which it is more probable to find pairs of particles. +This behavior is only observed at intermediate densities, and points to non-trivial behavior, which has yet to be fully studied. +A more systematic study of the two-point correlation function in the intermediate density regime is in progress, see\-~\cite{Ja23}. + +\begin{figure} + \hfil\includegraphics[width=7.5cm]{correlation0001.pdf} + \hfil\includegraphics[width=7.5cm]{correlation02.pdf}\par + \caption{The predictions of the two-point correlation function for the \eqformat{Simple Equation}, \eqformat{Medium Equation}, and \eqformat{Big Equation}, compared to a \eqformat{Quantum Monte Carlo} (QMC) simulation (computed by M.\-~Holzmann), for $\rho=0.0001$ and $\rho=0.02$ and $v=16e^{-|x|}$.} + \label{fig:2pt} +\end{figure} \section{Open problems}\label{sec:open} \indent -The Simplified Approach is thus a powerful tool to study the ground state of systems of interacting Bosons. -It reproduces the predictions of Bogolyubov theory, but goes far beyond: it is asymptotically correct at high densities, and very accurate at intermediate densities as well. -This tool may open the door to new attempts at proving some of the important outstanding problems concerning the Bose gas. +Lieb's Simplified approach is thus a powerful tool to study the ground state of systems of interacting Bosons. +It reproduces the predictions of Bogolyubov theory, but goes far beyond: it is asymptotically correct at high densities in addition to low ones, and is very accurate at intermediate densities as well. +We have discussed a number of results on the approach, but there still are many problems that remain open. +\bigskip + +\subsection{Connecting Lieb's Simplified approach to the many-body Bose gas} +\indent +For one, there is still no rigorous connection between Lieb's Simplified approach and the many-body Bose gas, other than the fact that the energy asymptotics for the Simple equation agree at low and high densities. +Even in that case, the computation of the energy for the Simple equation is independent from that for the many-body problem, and our understanding of the connection between the two models is still virtually non-existent. \bigskip \indent -There are two directions that could be investigated to make a rigorous connection between the Simplified Approach and the Bose gas. -One is to control the error made in the factorization assumption (see Assumption\-~\ref{assum:factorization}). -Another is to construct an approximation of the ground state wavefunction directly, for instance, in Bijl-Dingle-Jastrow form: +One way to approach this question would be to understand the approximation made in the factorization assumption. +This would require to bound the difference between the correlation functions $g_3,g_4$ and their factorized counterparts. +This may be doable for small enough densities, where inter-particle distances are large, and the interaction between particles mostly affects isolated clusters of particles. +One of the major difficulties of proceeding in this way is that it is hard to compute much about the ground state wavefunction, so establishing this clustering picture could be a challenge. +\bigskip + +\indent +Another idea (or, perhaps a first step) could be to use Lieb's Simplified approach to find an approximation for the ground state wavefunction itself. +As we have seen, the approach makes a prediction for the ground state energy, but not the wavefunction. +A natural candidate may be to construct such a wavefunction in Bijl-Dingle-Jastrow (BDJ) form: \begin{equation} \psi(x_1,\cdots,x_N)=\prod_{id$, then + \begin{equation} + |\partial^\alpha f|\in L_\infty(\mathbb R^d) + \end{equation} + for $|\alpha|\leqslant \ell$, in particular, if $p>\frac dm$, then + \nopagebreakaftereq + \begin{equation} + W_{m,p}(\mathbb R^d)\subset L_\infty(\mathbb R^d) + . + \end{equation} +\endtheo +\restorepagebreakaftereq +\bigskip + +\theo{Lemma}\label{lemma:Lq_from_Wmp} + If $f\in W_{m,p}(\mathbb R^d)$ with $mp\geqslant d$, then $f\in L_q(\mathbb R^d)$ for all $q\geqslant p$. +\endtheo +\bigskip + +\indent\underline{Proof}: + We apply the Sobolev Embedding Theorem\-~\ref{theo:sobolev} with $\ell=m-\frac dp$, so + \begin{equation} + (m-\ell)p=d + ,\quad + \frac1p-\frac md+\frac\ell d=0 + \end{equation} + and find that $f\in W_{\ell,q}(\mathbb R^d)\subset L_q(\mathbb R^d)$ for any $q\geqslant p$. +\qed +\bigskip + +\theoname{Theorem}{Young's inequality {\rm\cite[Section 4.2, Remark 2]{LL01}}}\label{theo:young} + If $f\in L_p(\Omega)$ and $g\in L_q(\Omega)$, then $f\ast g\in L_r(\Omega)$ with + \begin{equation} + 1+\frac1r=\frac1p+\frac 1q + \end{equation} + and + \nopagebreakaftereq + \begin{equation} + \|f\ast g\|_r\leqslant \|f\|_p\|g\|_q + . + \end{equation} +\endtheo +\restorepagebreakaftereq +\bigskip + +\subsection{Operators on Banach spaces} +\indent +To prove properties of the Simple Equation, we study the operator $K_e$, see\-~(\ref{Ke}), which acts on $W_{2,p}(\mathbb R^d)$. +Because $W_{2,p}(\mathbb R^d)$ is a Banach space (and only a Hilbert space for $p=2$), we will need some theorems on operators acting on Banach spaces as opposed to Hilbert spaces. +The tool we will use the most are ``generators of contraction semigroups'', which generalize the notion of positive definite self-adjoint operators on Hilbert spaces. +\bigskip + +\theo{Definition} + A contraction semigroup on a Banach space $X$ is a family $\{T(t),\ t\in[0,\infty)\}$ of bounded operators satisfying + \begin{equation} + T(0)=\mathds 1 + ,\quad + T(s)T(t)=T(s+t) + ,\quad + t\mapsto T(t)\varphi\mathrm{\ is\ continuous} + ,\quad + \|T(t)\|\leqslant 1 + \end{equation} + for any $s,t\in[0,\infty)$, $\varphi\in X$. +\endtheo +\bigskip + +\theo{Definition}\label{def:generate_semigroup} + An operator $A$ on a Banach space $X$ is said to generate a contraction semigroup if $A$ is closed, for any $\lambda>0$, $A+\lambda$ is invertible, and + \begin{equation} + \|(A+\lambda)^{-1}\|\leqslant\frac1\lambda + . + \end{equation} +\endtheo +\bigskip + +This is not the usual definition of the generator of a contraction semigroup, but it is equivalent to it, by the Hille-Yosida Theorem\-~\cite[Theorem X.47a]{RS75b}. +The Hille-Yosida Theorem also shows the following. +\bigskip + +\theoname{Theorem}{Hille-Yosida Theorem {\rm\cite[Theorem X.47a]{RS75b}}}\label{theo:hille_yosida} + If $A$ generates a contraction semigroup, then it is associated a unique contraction semigroup, denoted by $e^{-tA}$, and, for $\lambda>0$, + \nopagebreakaftereq + \begin{equation} + (A+\lambda)^{-1}=-\int_0^\infty dt\ e^{-\lambda t}e^{-tA} + . + \end{equation} +\endtheo +\restorepagebreakaftereq +\bigskip + +\theoname{Theorem}{Kato-Rellich Theorem on Banach space {\rm\cite[Theorem X.51]{RS75b}}}\label{theo:kato_rellich} + Let $A$ and $B$ generate contraction semigroups on a Banach space, such that the domain $\mathcal D(A)$ of $A$ is a subset of the domain $\mathcal D(B)$ of $B$. + If, for $\varphi\in\mathcal D(A)$, + \begin{equation} + \|B\varphi\|\leqslant a\|A\varphi\|+b\|\varphi\| + \end{equation} + for some $b\geqslant 0$ and $a\in[0,1)$, then $A+B$ generates a contraction semigroup. +\endtheo +\bigskip + +Note that this theorem holds under the more general condition that $B$ is ``accretive'', see\-~\cite[p.\-~240]{RS75b}, but this less general formulation is sufficient for our purposes. +\bigskip + +\theoname{Theorem}{Trotter product formula {\rm\cite[Theorem X.51]{RS75b}}}\label{theo:trotter} + Given two operators $A$ and $B$ on a Banach space such that $A$, $B$ and $A+B$ generate contraction semigroups, then for $t\geqslant 0$, + \nopagebreakaftereq + \begin{equation} + e^{-t(A+B)} + = + \lim_{N\to\infty}\left(e^{-t\frac 1N A}e^{-t\frac1N B}\right)^N + . + \end{equation} +\endtheo +\restorepagebreakaftereq +\bigskip + +\indent +We will also use the Hilbert space version of this theorem, stated below. +\bigskip + +\theoname{Theorem}{Trotter product formula on a Hilbert space {\rm\cite[Theorem 5.12]{Te14}}}\label{theo:trotter_hilbert} + Given two operators $A$ and $B$ on a Hilbert space such that $A$, $B$ and $A+B$ are self-adjoint, and bounded from below, then for $t\geqslant 0$, + \nopagebreakaftereq + \begin{equation} + e^{-t(A+B)} + = + \lim_{N\to\infty}\left(e^{-t\frac 1N A}e^{-t\frac1N B}\right)^N + . + \end{equation} +\endtheo +\restorepagebreakaftereq +\bigskip + +The connection to Theorem\-~\ref{theo:trotter} is that a self-adjoint operator on a Hilbert space generates a contraction semigroup if and only if its spectrum is $>\epsilon>0$ (this follows from the spectral theorem). +One can then extend the result to operators that are bounded below by adding a multiple of the identity to raise the spectrum above $\epsilon$. +\bigskip + +\theoname{Theorem}{Heat kernel}\label{theo:heat_contractive} + The operator $-\Delta$ from $W_{2,p}(\mathbb R^d)$ to $L_p(\mathbb R^d)$ generates a contractive semigroup. +\endtheo +\bigskip + +\indent\underline{Proof}: + The Laplacian is trivially bounded on $W_{2,p}(\mathbb R^d)$, so it is closed. + Furthermore, by Theorem\-~\ref{theo:yukawa}, for $\lambda>0$, $(-\Delta+\lambda)$ is invertible and, for $\psi\in L_p(\mathbb R^d)$, + \begin{equation} + (-\Delta+\lambda)^{-1}\psi=Y_{\sqrt\lambda}\ast\psi + \end{equation} + so, by Young's inequality (Theorem\-~\ref{theo:young}), + \nopagebreakaftereq + \begin{equation} + \|(-\Delta+\lambda)^{-1}\|\leqslant\|Y_{\sqrt\lambda}\|_1=\frac1\lambda + . + \end{equation} +\qed +\restorepagebreakaftereq + +\subsection{Fourier analysis} +\theoname{Definition}{Fourier transform} + The Fourier transform of a function $f$ is + \begin{equation} + \hat f(k):=\int dx\ e^{ikx}f(x) + \label{fourier} + \end{equation} + which is defined for any $f$ such that this integral is finite. +\endtheo +\bigskip + +\theoname{Theorem}{{\rm\cite[Theorems 5.3, 5.5]{LL01}}}\label{theo:fourierinv} + If $f\in L_2(\mathbb R^d)$, then $\hat f$ is well defined, and + \begin{equation} + f(x)=\int\frac{dk}{(2\pi)^d}e^{-ikx}\hat f(k) + . + \label{fourierinv} + \end{equation} + In addition, $\hat f\in L_2(\mathbb R^d)$ and + \nopagebreakaftereq + \begin{equation} + \|\hat f\|_2=\|f\|_2 + . + \end{equation} +\endtheo +\restorepagebreakaftereq + +%\theoname{Theorem}{Hausdorff-Young inequality {\rm\cite[Theorem 1]{Be75}}}\label{theo:hausdorff_young} +% For $p\in(1,2]$ and $f\in L_p(\mathbb R^d)$, then $\hat f$ is well-defined and +% \begin{equation} +% \|\hat f\|_{p'}\leqslant C_p^d\|f\|_p +% ,\quad +% \frac1{p'}:=1-\frac1p +% ,\quad +% C_p:=\sqrt{(p^{\frac1p}(p')^{-\frac1{p'}})} +% . +% \end{equation} +% In particular, $\hat f\in L_{p'}(\mathbb R^d)$. +%\endtheo +%\bigskip +% +%\theo{Corollary}\label{cor:hausdorff_young} +% For $q\geqslant 2$, $f\in L_2(\mathbb R^d)$ such that $\hat f\in L_{q'}(\mathbb R^d)$, then +% \nopagebreakaftereq +% \begin{equation} +% \|f\|_{q}\leqslant \frac{C_{q'}^d}{(2\pi)^d}\|\hat f\|_{q'} +% ,\quad +% \frac1{q'}:=1-\frac1q +% . +% \end{equation} +%\endtheo +%\restorepagebreakaftereq +%\bigskip +% +%\indent\underline{Proof}: +% The idea is to apply Theorem\-~\ref{theo:hausdorff_young} to the inverse Fourier transform of $f$. +% First, of all, the inverse Fourier transform of $f$ exists by Theorem\-~\ref{theo:fourierinv} because $f\in L_2(\mathbb R^d)$. +% Let +% \begin{equation} +% \check f(k):=\int\frac{dx}{(2\pi)^d}e^{-ikx}f(x)\equiv \frac1{(2\pi)^d}\hat f(-k) +% . +% \end{equation} +% By Theorem\-~\ref{theo:hausdorff_young} (noting that $\check f\in L_{q'}(\mathbb R^d)$ if and only if $\hat f\in L_{q'}(\mathbb R^d)$ and that $q\geqslant 2$ if and only if $q'\in(1,2]$), +% \begin{equation} +% \|f\|_q=\|\widehat{\check f}\|_q +% \leqslant C_{q'}^d\|\check f\|_{q'} +% =\frac{C_{q'}^d}{(2\pi)^d}\|\hat f\|_{q'} +% . +% \end{equation} +%\qed + +\subsection{Positivity preserving operators on Banach spaces}\label{app:positivity_preserving} \theoname{Definition}{Positivity preserving operators}\label{def:positivity_preserving} An operator $A$ from a Banach space of real-valued functions $\mathcal B_1$ to another $\mathcal B_2$ is said to be {\it positivity preserving} if, for any $f\in\mathcal B_1$ such that $f(x)\geqslant0$, $Af(x)\geqslant 0$. \endtheo \bigskip -\theo{Lemma}\label{lemma:add_inv} - If $\mathrm{spec}(A+B)>\epsilon>0$, and $e^{-tA}$ and $e^{-t B}$ are positivity preserving for all $t>0$, then for all $t>0$, $e^{-t(A+B)}$ and $(A+B)^{-1}$ are positivity preserving. +\theo{Theorem}\label{theo:add_inv} + If $A$, $B$, and $A+B$ generate contraction semigroups, and $e^{-tA}$ and $e^{-t B}$ are positivity preserving for all $t\geqslant 0$, then for all $t\geqslant 0$, $e^{-t(A+B)}$ and $(A+B+\lambda)^{-1}$ are positivity preserving for all $\lambda>0$. \endtheo \bigskip \indent\underline{Proof}: - We write + By the Hille-Yosida Theorem\-~\ref{theo:hille_yosida}, \begin{equation} - (A^{-1}+B)^{-1} + (A+B+\lambda)^{-1} = - \int_0^\infty dt\ e^{-t(A^{-1}+B)} + \int_0^\infty dt\ e^{-t\lambda}e^{-t(A+B)} \end{equation} - indeed, by the spectral theorem, + and by the Trotter product formula (Theorem\-~\ref{theo:trotter}), \begin{equation} - \int_0^\infty dt\ e^{-t(A^{-1}+B)} + e^{-t(A+B)} = - \int_0^\infty dt\ \int_{\mathrm{spec}(A^{-1}+B)}d\sigma\ e^{-t\sigma}P_\sigma - = - \int_{\mathrm{spec}(A^{-1}+B)}d\sigma\ \frac1\sigma P_\sigma - =(A^{-1}+B)^{-1} - . - \end{equation} - By the Trotter product formula (Theorem\-~\ref{theo:trotter}), - \begin{equation} - e^{-t(A^{-1}+B)} - = - \lim_{N\to\infty}(e^{-\frac tNA^{-1}}e^{-\frac tNB})^N + \lim_{N\to\infty}(e^{-\frac tNA}e^{-\frac tNB})^N . \end{equation} + We conclude the proof by noting that the product of positivity preserving operators is positivity preserving, and that the set of positivity preserving operators is closed. \qed \bigskip -\theoname{Theorem}{Positivity of the heat kernel}\label{theo:heat} - Given $t>0$, the operator $e^{t\Delta}$ from $L_2(\mathbb R^d)$ to $L_2(\mathbb R^d)$ is positivity preserving. +\theoname{Lemma}{Positivity of the heat kernel on $L_{2}(\mathbb R^d)$}\label{lemma:heat_L2} + Given $t>0$, the operator $e^{t\Delta}$ on $L_2(\mathbb R^d)$ is positivity preserving. \endtheo \bigskip \indent\underline{Proof}: - If + Given $\psi\in L_2(\mathbb R^d)$, by Theorem\-~\ref{theo:fourierinv}, the Fourier transform of $\psi$ is $\hat\psi\in L_2(\mathbb R^d)$, and \begin{equation} - \psi_t(x)=e^{t\Delta}\psi_0(x) + \int dx\ e^{ikx}e^{t\Delta}\psi + = + e^{-tk^2}\hat\psi(k) \end{equation} - then + which is in $L_2(\mathbb R^d)$, so we can take an inverse Fourier transform: \begin{equation} - \partial_t\psi_t(x)=\Delta\psi_t(x) - . - \end{equation} - The Fourier transform of $\psi_t$ is defined as - \begin{equation} - \hat\psi_t(k):=\int dx\ e^{ikx}\psi_t(x) - \end{equation} - which satisfies - \begin{equation} - \partial_t\hat\psi_t(k)=-k^2\hat\psi(k) - \end{equation} - so - \begin{equation} - \hat\psi_t(k)=e^{-tk^2}\hat\psi_0(k) - \end{equation} - thus, taking an inverse Fourier transform, - \begin{equation} - \psi_t(x)=\left(\int\frac{dk}{(2\pi)^d}\ e^{-ikx}e^{-tk^2}\right)\ast\psi_0(x) + e^{t\Delta}\psi(x)=\left(\int\frac{dk}{(2\pi)^d}\ e^{-ikx}e^{-tk^2}\right)\ast\psi(x) \end{equation} + (to see that the convolution appears, one can do a simple explicit computation, or use\-~\cite[Theorem\-~5.8]{LL01}) and + \nopagebreakaftereq \begin{equation} \int\frac{dk}{(2\pi)^d}\ e^{-ikx}e^{-tk^2} =\prod_{i=1}^d\int_{-\infty}^\infty\frac{dk_i}{2\pi}\ e^{-ik_ix_i}e^{-tk_i^2} @@ -2412,45 +3578,103 @@ In this appendix we gather a few useful definitions and results from functional . \end{equation} \qed +\restorepagebreakaftereq \bigskip -\theo{Theorem}\label{theo:schrodinger} - Given $t>0$ and a function $v$, the operator $e^{-t(-\Delta+v)}$ is positivity preserving. - In addition, if $\mathrm{spec}(-\Delta+v)>\epsilon>0$, then $(-\Delta+v)^{-1}$ is positivity preserving. +\theoname{Theorem}{Positivity of the heat kernel on $W_{2,p}(\mathbb R^d)$}\label{theo:heat} + Given $t>0$, the operator $e^{t\Delta}$ on $W_{2,p}(\mathbb R^d)$ with $p\geqslant \min\{2,\frac d2\}$ is positivity preserving. \endtheo \bigskip \indent\underline{Proof}: - By the Trotter product formula (Theorem\-~\ref{theo:trotter}), + We take $\psi\in W_{2,p}(\mathbb R^d)$. + If $p\leqslant 2$, then, + %, since + %\begin{equation} + % \frac12\geqslant\frac1p-\frac2d + %\end{equation} + by the Sobolev Embedding Theorem\-~\ref{theo:sobolev} (more precisely, by Lemma\-~\ref{lemma:Lq_from_Wmp}), we have $\psi\in L_2(\mathbb R^d)$. + Therefore, if $\psi\geqslant 0$, then $e^{t\Delta}\psi\geqslant 0$ by Lemma\-~\ref{lemma:heat_L2}. + \bigskip + + \indent + Let us now turn to $p>2$. + We introduce a smooth cutoff function $\chi_N$ which is $\mathcal C_\infty$ (infinitely continuously differentiable), non-negative, and is equal to $1$ for $|x|\leqslant N$ and equal to $0$ for $|x|\geqslant N+1$. + For every $N\geqslant 0$, $\chi_N\psi\in W_{2,p}(\mathbb R^d)\cap L_2(\mathbb R^d)$ (see, e.g. \cite[Section\-~6.4]{LL01}). + Therefore, if $\psi\geqslant 0$, then by Lemma\-~\ref{lemma:heat_L2}, $e^{t\Delta}\chi_N\psi\geqslant 0$. + Furthermore, since $e^{t\Delta}$ is bounded (by Theorem\-~\ref{theo:heat_contractive}), it is continuous, and \begin{equation} - e^{-t(-\Delta+v)} - = - \lim_{N\to\infty}(e^{\frac tN\Delta}e^{-\frac tNv})^N + \lim_{N\to\infty}\|\psi-\chi_N\psi\|_{W_{2,p}(\mathbb R^d)}=0 \end{equation} - which is positivity preserving, because $e^{-\frac tNv}$ is a multiplication operator by a non-negative function. - The second point follows from lemma\-~\ref{lemma:add_inv}, in which we take $A=-\Delta$ and $B=v$ ($e^{-tv}$ is a multiplication operator by a non-negative function, which is positivity preserving, and so is $e^{t\Delta}$, by Theorem\-~\ref{theo:heat}). + so $e^{t\Delta}\psi\geqslant 0$. \qed \bigskip -{\bf Remark}: In particular, taking the potential to be $\eta+v(x)$, this implies that $e^{-t(-\Delta+v(x)+\eta)}$ and $(-\Delta+v(x)+\eta)^{-1}$ are positivity preserving for any $\eta,v(x)\geqslant 0$. -\bigskip -\theo{Lemma}\label{lemma:conv} - If $A$ is positivity preserving, $f\in L_1(\mathbb R^d)$ such that $f\geqslant 0$ and $\mathrm{spec}(A-f\ast)>\epsilon>0$ and $e^{-tA}$ is positivity preserving, then $(A-f\ast)^{-1}$ is positivity preserving. +%\theo{Theorem}\label{theo:schrodinger} +% Given $t>0$ and a function $v$, the operator $e^{-t(-\Delta+v)}$ is positivity preserving. +% In addition, if $\mathrm{spec}(-\Delta+v)>\epsilon>0$, then $(-\Delta+v)^{-1}$ is positivity preserving. +%\endtheo +%\bigskip +% +%\indent\underline{Proof}: +% By the Trotter product formula (Theorem\-~\ref{theo:trotter}), +% \begin{equation} +% e^{-t(-\Delta+v)} +% = +% \lim_{N\to\infty}(e^{\frac tN\Delta}e^{-\frac tNv})^N +% \end{equation} +% which is positivity preserving, because $e^{-\frac tNv}$ is a multiplication operator by a non-negative function. +% The second point follows from theorem\-~\ref{theo:add_inv}, in which we take $A=-\Delta$ and $B=v$ ($e^{-tv}$ is a multiplication operator by a non-negative function, which is positivity preserving, and so is $e^{t\Delta}$, by Theorem\-~\ref{theo:heat}). +%\qed +%\bigskip +% +%{\bf Remark}: In particular, taking the potential to be $\eta+v(x)$, this implies that $e^{-t(-\Delta+v(x)+\eta)}$ and $(-\Delta+v(x)+\eta)^{-1}$ are positivity preserving for any $\eta,v(x)\geqslant 0$. +%\bigskip +% +%\theo{Lemma}\label{lemma:conv} +% If $A$ is positivity preserving, $f\in L_1(\mathbb R^d)$ such that $f\geqslant 0$ and $\mathrm{spec}(A-f\ast)>\epsilon>0$ and $e^{-tA}$ is positivity preserving, then $(A-f\ast)^{-1}$ is positivity preserving. +%\endtheo +%\bigskip +% +%\indent\underline{Proof}: +% This follows from theorem\-~\ref{theo:add_inv}, noting that $e^{tf\ast}$ is obviously positivity preserving. +%\qed +%\bigskip +% +%{\bf Remark}: +%In particular, if $v(x),\eta> 0$, $\|f\|_1\leqslant\eta$, then $(-\Delta+v+\eta-f\ast)^{-1}$ is positivity preserving. +%The assumption $\|f\|_1\leqslant\eta$ ensures that $-\Delta+v+\eta-f\ast$ has a spectrum that is bounded away from 0. + +\subsection{The Yukawa potential} +\theoname{Theorem}{{\rm\cite[Theorem 6.23]{LL01}}}\label{theo:yukawa} + For $d\geqslant 1$, $\mu>0$, we define the Yukawa potential + \begin{equation} + Y_\mu(x):=\int_0^\infty\frac{dt}{(4\pi t)^{\frac n2}}\ e^{-\frac{|x|^2}{4t}-\mu^2t} + \end{equation} + satisfies + \begin{equation} + \|Y_\mu\|_1=\frac1{\mu^2} + \end{equation} + is in $L_q(\mathbb R^d)$ for + \begin{equation} + \left\{\begin{array}{ll} + q\in[1,\infty]&\mathrm{if\ }d=1\\ + q\in[1,\infty)&\mathrm{if\ }d=2\\ + q\in[1,\frac d{d-2}]&\mathrm{if\ }d\geqslant 3 + \end{array}\right. + \end{equation} + and, for any $f\in L_p(\mathbb F^d)$, $p\in[1,\infty]$, + \nopagebreakaftereq + \begin{equation} + (-\Delta+\mu^2)(Y_\mu\ast f)=f + . + \end{equation} \endtheo -\bigskip - -\indent\underline{Proof}: - This follows from lemma\-~\ref{lemma:add_inv}, noting that $e^{tf\ast}$ is obviously positivity preserving. -\qed -\bigskip - -{\bf Remark}: -In particular, if $v(x),\eta> 0$, $\|f\|_1\leqslant\eta$, then $(-\Delta+v+\eta-f\ast)^{-1}$ is positivity preserving. -The assumption $\|f\|_1\leqslant\eta$ ensures that $-\Delta+v+\eta-f\ast$ has a spectrum that is bounded away from 0. +\restorepagebreakaftereq \subsection{The Perron-Frobenius theorem} -\theoname{Theorem}{(Generalized) Perron-Frobenius theorem}\label{theo:perron_frobenius} +\theoname{Theorem}{(Generalized) Perron-Frobenius theorem {\rm\cite[Theorem 1.1]{Du06}}}\label{theo:perron_frobenius} Let $A$ be a compact operator from a Banach space of real-valued functions $\mathcal B_1$ to another $\mathcal B_2$. If $A$ is positivity preserving (see Definition\-~\ref{def:positivity_preserving}) and has a finite and positive spectral radius $r(A)$ ($r(A):=\sup_{\lambda\in\mathrm{spec}(A)}|\lambda|$), then $r(A)$ is a non-degenerate eigenvalue whose corresponding eigenvector $f$ is non-negative: \nopagebreakaftereq @@ -2463,6 +3687,8 @@ The assumption $\|f\|_1\leqslant\eta$ ensures that $-\Delta+v+\eta-f\ast$ has a \endtheo \restorepagebreakaftereq +(The original Perron-Frobenius theorem applies to matrices; this generalized version is also called the Krein-Rutman theorem.) + \section{Elements of harmonic analysis}\label{app:harmonic} \indent In this appendix, we state some useful results from harmonic analysis. @@ -2487,11 +3713,197 @@ In this appendix, we state some useful results from harmonic analysis. \restorepagebreakaftereq \bigskip -\theoname{Theorem}{\cite[Theorem 9.4]{LL01}}\label{theo:harmonic} +\theoname{Theorem}{{\rm\cite[Theorem 9.4]{LL01}}}\label{theo:harmonic} A function that is subharmonic on $A$ achieves its maximum on the boundary of $A$. A function that is superharmonic on $A$ achieves its minimum on the boundary of $A$. \endtheo +\section{The operator $K_e$}\label{app:Ke} +\indent +In this appendix we prove some properties on the operator $K_e$, formally defined as +\begin{equation} + K_e:=\left(-\frac1m\Delta+v+4e\right)^{-1} +\end{equation} +from $L_p(\mathbb R^d)$ to the Sobolev space $W_{2,p}(\mathbb R^d)$, see Section\-~\ref{app:sobolev}. +In particular, we will prove that this operator is well-defined and positivity-preserving. +The first step is to prove that $v$ is relatively bounded with respect to $-\frac1m\Delta+4e$. +\bigskip + +\theo{Lemma}\label{lemma:v_relative_bound} + For $p\geqslant \min\{\frac d2,1\}$ and $f\in W_{2,p}(\mathbb R^d)$, if $v\in L_r(\mathbb R^d)$ with $r\geqslant p$, and $e,m>0$, there exists $b>0$ and $a\in[0,1)$ such that + \begin{equation} + \|vf\|_p\leqslant a\|(-{\textstyle \frac1m}\Delta+4e)f\|_p+b\|f\|_p + \end{equation} + in other words, $v$ is relatively bounded with respect to $-\frac1m\Delta+4e$. +\endtheo +\bigskip + +\indent\underline{Proof}: + First of all, we split $v$ into the sum of a bounded function and a function with small $L_r$ norm: + \begin{equation} + v=v_\infty+v_r + ,\quad + v_\infty:=v\mathds 1_{v\leqslant b} + ,\quad + v_r:=v\mathds 1_{v>b} + \end{equation} + where $\mathds 1_{v\leqslant b}\in\{0,1\}$ is equal to 1 if and only if $v\leqslant b$ and similarly for $\mathds 1_{v>b}$. + We then bound + \begin{equation} + \|vf\|_p\leqslant\|v_\infty f\|_p+\|v_rf\|_p + . + \end{equation} + Since $|v_\infty|\leqslant b$, + \begin{equation} + \|v_\infty f\|_p\leqslant b\|f\|_p + . + \end{equation} + Next, by the Sobolev Embedding Theorem\-~\ref{theo:sobolev} (or rather by Lemma\-~\ref{lemma:Lq_from_Wmp}), since $p\geqslant\frac d2$, $f\in L_q(\mathbb R^d)$ for any $q\geqslant p$. + Therefore, by the H\"older inequality (Theorem\-~\ref{theo:holder}), + \begin{equation} + \|v_rf\|_p\leqslant\|v_r\|_{r}\|f\|_q + ,\quad + \frac1q=\frac1p-\frac1r + \end{equation} + (note that $q\in[1,\infty]$). + Furthermore, by Theorem\-~\ref{theo:yukawa}, + \begin{equation} + f=(-\Delta+4em)(Y_{2\sqrt{em}}\ast f)=Y_{2\sqrt{em}}\ast(-\Delta+4em)f + \end{equation} + and by Young's inequality (Theorem\-~\ref{theo:young}), + \begin{equation} + \|f\|_q\leqslant \|Y_{2\sqrt{em}}\|_{\frac r{r-1}}\|(-\Delta+4em)f\|_p + . + \end{equation} + Note that, if $d<3$ by Theorem\-~\ref{theo:yukawa}, $Y_{2\sqrt{em}}\in L_{\frac r{r-1}}(\mathbb R^d)$ for any $r$, and if $d\geqslant 3$, $Y_{2\sqrt{em}}\in L_l(\mathbb R^d)$ for $l\leqslant\frac d{d-2}$, and $\frac r{r-1}\leqslant \frac d{d-2}$ because $r\geqslant \frac d2$, so $Y_{2\sqrt{em}}\in L_{\frac r{r-1}}(\mathbb R^d)$. + Thus, + \begin{equation} + \|vf\|_p\leqslant b\|f\|_p+m\|Y_{2\sqrt{em}}\|_{\frac r{r-1}}\|v_r\|_r\|(-{\textstyle\frac1m}\Delta+4e)f\|_p + . + \end{equation} + Finally, by dominated convergence, + \begin{equation} + \lim_{b\to\infty}\|v_r\|_r=0 + \end{equation} + so, taking $b$ to be large enough, + \nopagebreakaftereq + \begin{equation} + m\|Y_{2\sqrt{em}}\|_{\frac r{r-1}}\|v_r\|_r<1 + . + \end{equation} + %We wish to apply the Hausdorff-Young inequality to relate $\|f\|_q$ to a norm of $\hat f$. + %We will use Corollary\-~\ref{cor:hausdorff_young}, for which we must first check that $f\in L_2(\mathbb R^d)$. + %This follows from the Sobolev embedding Theorem\-~\ref{theo:sobolev} using the fact that $f\in W_{2,p}(\mathbb R^d)$ and + %\begin{equation} + % \frac1p<\frac2d<\frac12+\frac2d + % . + %\end{equation} + %We then apply Corollary\-~\ref{cor:hausdorff_young}: + %\begin{equation} + % \|f\|_2 + %\end{equation} + %To apply the Hausdorff-Young inequality, we also need $q\geqslant 2$, we we can achieve if we choose $r$ such that + %\begin{equation} + % \frac1r\geqslant\frac1p-\frac12 + % . + %\end{equation} + %We then apply the Hausdorff-Young inequality, Theorem\-~\ref{theo:hausdorff_young}, to the inverse Fourier transform of $f$: + %\begin{equation} + % \|f\|_q + % \leqslant C_{q'}\left\|\int\frac{dk}{(2\pi)^d}e^{-ikx}f(k)\right\|_{q'} + % =(2\pi)^{-d}C_{q'}\|\hat f\|_{q'} + % ,\quad + % \frac1{q'}=1-\frac1q + % . + %\end{equation} +\qed +\restorepagebreakaftereq +\bigskip + +\theo{Lemma}\label{lemma:Ke_pos} + If $v\geqslant 0$, $v\in L_p(\mathbb R^d)$ with $p\geqslant 1$ and $p>\frac d2$, the operator $-\frac1m\Delta+v+4e$ from $W_{2,p}(\mathbb R^d)$ to $L_p(\mathbb R^d)$ is invertible and its inverse is bounded and positivity preserving. +\endtheo +\bigskip + +\indent\underline{Proof}: + We check the assumptions of the Kato-Rellich Theorem\-~\ref{theo:kato_rellich}. + By Theorem\-~\ref{theo:heat_contractive}, $-\frac1m\Delta$, defined on $W_{2,p}(\mathbb R^d)$, generates a contractive semigroup. + In addition, the domain of $v$ is $L_\infty(\mathbb R^d)$: indeed, by the H\"older inequality (Theorem\-~\ref{theo:holder}), for $\psi\in L_\infty(\mathbb R^d)$, + \begin{equation} + \|v\psi\|_p\leqslant\|v\|_p\|\psi\|_\infty + . + \end{equation} + Furthermore, by the Sobolev Embedding Theorem\-~\ref{theo:sobolev}, since $p>\frac d2$, $W_{2,p}(\mathbb R^d)\subset L_\infty(\mathbb R^d)$. + In addition, since $v\geqslant 0$, it generates the contraction semigroup $e^{-tv(x)}$. + Thus, by Lemma\-~\ref{lemma:v_relative_bound} and the Kato-Rellich Theorem\-~\ref{theo:kato_rellich}, $(-\frac 1m\Delta+v)$ generates a contraction semigroup. + By Definition\-~\ref{def:generate_semigroup}, $(-\frac1m\Delta+v+4e)$ is invertible and its inverse is bounded. + \bigskip + + \indent + By Theorem\-~\ref{theo:heat}, $e^{\frac1mt\Delta}$ is positivity preserving, as $p>\frac d2$. + In addition, $e^{-tv}$ is positivity preserving because $v\geqslant 0$. + Therefore, by Theorem\-~\ref{theo:add_inv}, $(-\frac1m\Delta+v+4e)^{-1}$ is positivity preserving. +\qed +\bigskip + +\indent +Thus, $K_e$ is defined as an operator from $W_{2,p}(\mathbb R^d)$ to $L_p(\mathbb R^d)$. +As we will now prove, $K_e\psi$ is actually also defined for $\psi\in L_1(\mathbb R^d)$. +\bigskip + +\theo{Lemma}\label{lemma:Ke_extend} + $K_e$ extends to a bounded operator from $L_q(\mathbb R^d)$ to $L_q(\mathbb R^d)$ for any $q\geqslant 1$. +\endtheo +\bigskip + +\indent\underline{Proof}: + We have + \begin{equation} + (-{\textstyle\frac 1m}\Delta+4e+v)^{-1} + = + (-{\textstyle\frac 1m}\Delta+4e)^{-1} + - + (-{\textstyle\frac 1m}\Delta+4e)^{-1} + v + (-{\textstyle\frac 1m}\Delta+4e+v)^{-1} + \end{equation} + (this identity is called the ``second resolvent identity'', and follows from an elementary computation), which we rewrite using Theorem\-~\ref{theo:yukawa}: + \begin{equation} + K_e\psi + =\frac1mY_{2\sqrt{me}}\ast\psi + -\frac1m Y_{2\sqrt{me}}\ast(vK_e\psi) + . + \end{equation} + Writing + \begin{equation} + \psi(x)=\psi_+(x)-\psi_-(x) + ,\quad + \psi_+(x):=\mathds 1_{\psi(x)\geqslant 0}\psi(x)\geqslant 0 + ,\quad + \psi_-(x):=-\mathds 1_{\psi(x)<0}\psi(x)\geqslant 0 + \end{equation} + we have, formally, for $q\geqslant 1$, + \begin{equation} + \|K_e\psi\|_q\leqslant \|K_e\psi_+\|_q+\|K_e\psi_-\|_q + . + \end{equation} + Now, since $Y_{2\sqrt{me}}\geqslant 0$, $v\geqslant 0$ and $K_e$ is positivity preserving and $\psi_\pm(x)\geqslant 0$, + \begin{equation} + 0\leqslant K_e\psi_\pm + \leqslant\frac1mY_{2\sqrt{me}}\ast\psi_\pm + \end{equation} + so, by Young's inequality (Theorem\-~\ref{theo:young}), + \nopagebreakaftereq + \begin{equation} + \|K_e\psi\|_q + \leqslant \frac 1m\|Y_{2{\sqrt{me}}}\|_1(\|\psi_+\|_q+\|\psi_-\|_q) + \leqslant \frac 2{m^2e}\|\psi\|_q + . + \end{equation} +\qed +\restorepagebreakaftereq + + \section{Proof of Theorem \expandonce{\ref{theo:compleq}}}\label{app:proof_factorization} \subsection{Factorization} @@ -2504,11 +3916,11 @@ We need to compute these up to order $V^{-2}$, because one of the terms in the e \theo{Lemma}\label{lemma:g3} Assumption\-~\ref{assum:factorization} with $i=3$ and\-~(\ref{cd_g3g4}) imply that \begin{equation} - g_3(x,y,z)=(1-u_3(x-y))(1-u_3(x-z))(1-u_3(y-z))(1+O(V^{-2})) + g_3(x,y,z)=(1-u_3(x-y))(1-u_3(x-z))(1-u_3(y-z)) \end{equation} with \begin{equation} - u_3(x-y):=u(x-y)+\frac{w_3(x-y)}V + u_3(x-y)=\left(u(x-y)+\frac{w_3(x-y)}V\right)(1+O(V^{-2})) \label{u3} \end{equation} \nopagebreakaftereq @@ -2564,7 +3976,7 @@ We need to compute these up to order $V^{-2}$, because one of the terms in the e \right) . \end{equation} - By\-~(\ref{3V}), + Dividing by the integral, we find that, by\-~(\ref{3V}), \begin{equation} \begin{largearray} (1-u_3(x-y)) @@ -2585,12 +3997,14 @@ We need to compute these up to order $V^{-2}$, because one of the terms in the e \label{intu0} \end{equation} Thus, + \nopagebreakaftereq \begin{equation} 1-u_3(x-y)=(1-u(x-y)) \left(1-\int\frac{dz}{V}\ u(x-z)u(y-z)+O(V^{-2})\right) . \end{equation} \qed +\restorepagebreakaftereq \subsubsection{Factorization of $g_4$} @@ -2599,11 +4013,11 @@ We need to compute these up to order $V^{-2}$, because one of the terms in the e \begin{equation} g_4(x_1,x_2,x_3,x_2)= \left(\prod_{i\displaystyle l} @@ -2787,10 +4204,6 @@ with \end{equation} and by\-~(\ref{w3}), -%break to avoid big white spaces -\vfill -\eject - \begin{equation} \begin{array}{r@{\ }>\displaystyle l} \bar G_4(x-y) @@ -2859,6 +4272,66 @@ and, by\-~(\ref{intu0}) and\-~(\ref{Pi}), Inserting these into\-~(\ref{g2bar}), we find\-~(\ref{compleq}). \qed +\section{The momentum distribution in Bogolyubov theory}\label{app:bog_Nk} +\indent +In this appendix we recover the prediction of Bogolyubov theory for the momentum distribution\-~(\ref{momentum_distribution_def}) from\-~\cite[Appendix\-~A]{LSe05}. +\bigskip + +\indent +The expression of the Bogolyubov Hamiltonian in second quantized form is\-~\cite[(A.19)-(A.20)]{LSe05} +\begin{equation} + H_B=\frac12N\rho\hat v(0) + +\sum_{k\neq0}\left( + (\epsilon(k)+\rho\hat v(k))a_k^\dagger a_k+\frac12\rho \hat v(k)(a_ka_{-k}+a_{-k}^\dagger a_k^\dagger + \right) +\end{equation} +where $\epsilon(k)$ is given below\-~\cite[(A.6)]{LSe05}: +\begin{equation} + \epsilon(k)=\frac{k^2}{2m} + \label{epsilon} +\end{equation} +(recall that we are using units where $\hbar=1$). +Without going through the details of the second quantized formalism, suffice it to say that $a_k^\dagger a_k$ is an operator that counts the number of particles in the momentum $k$ state. +Therefore, the average number of particles in the momentum $k$ state is +\begin{equation} + N_k=\left<\psi_0\right|a_k^\dagger a_k\left|\psi_0\right> + =\partial_{\epsilon(k)}E_0 + . +\end{equation} +The momentum distribution is the probability of finding a particle in the momentum state $k$, and so, +\begin{equation} + \mathcal M_0^{(\mathrm{Bogolyubov})}(k)=\frac1\rho N_k=\frac1\rho\partial_{\epsilon(k)}E_0 +\end{equation} +The ground state energy $E_0$ can be computed exactly\-~\cite[(A.26)]{LSe05}: +\begin{equation} + E_0=\frac12N\rho\hat v(0) + -\frac12\sum_k \left(\epsilon(k)+\rho\hat v(k)-\sqrt{\epsilon(k)^2+2\epsilon(k)\rho\hat v(k)}\right) +\end{equation} +(we obtain this from\-~\cite[(A.26)]{LSe05} by first writing the integral over $|k|$ as one over $k$: $\int dk=4\pi\int_0^\infty d|k||k|^2$, and then undoing the ``bulk limit'' by replacing $\int dk$ by $\frac{8\pi^3}V\sum_k$). +Therefore, +\begin{equation} + \mathcal M_0^{(\mathrm{Bogolyubov})}(k)=\frac1\rho\partial_{\epsilon(k)}E_0 + =-\frac1{2\rho}\left( + 1-\frac{\epsilon(k)+\rho\hat v(k)}{\sqrt{\epsilon(k)^2+2\epsilon(k)\rho\hat v(k)}} + \right) +\end{equation} +and so, by\-~(\ref{epsilon}), +\begin{equation} + \mathcal M_0^{(\mathrm{Bogolyubov})}(k) + =\frac1{2\rho}\left( + \frac{k^2+2m\rho\hat v(k)}{\sqrt{k^4+4mk^2\rho\hat v(k)}}-1 + \right) + . +\end{equation} +This is not quite\-~(\ref{Nk_conj}), which we may rewrite using\-~(\ref{kappak_conj}) as +\begin{equation} + \frac1{2\rho}\left(\frac{k^2+8m\pi\rho a}{\sqrt{k^4+16m\pi k^2\rho a}}-1\right) + . +\end{equation} +To get this expression, we must do one more operation, which is required to get correct low-density predictions in Bogolyubov theory, which is to replace $v$ with a ``pseudo-potential''\-~\cite{LHY57}, which is defined as $4\pi a\delta(x)$. +In other, words, we replace $\hat v(k)$ with the constant $4\pi a$. +Thus, $\mathcal M_0^{(\mathrm{Bogolyubov})}$ becomes\-~(\ref{Nk_conj}). + \vfill \eject @@ -2910,7 +4383,7 @@ Prove that the solution of\-~(\ref{scat_softcore_spherical}) is\-~(\ref{sol_soft Prove that the ground state $\psi_0$ of\-~(\ref{Ham}) is unique and $\psi_0\geqslant 0$.\par {\it Hint}: Instead of the Hamiltonian $H_N$, consider the operator $e^{-tH_N}$ for $t\geqslant0$. -Check that Theorem\-~\ref{theo:schrodinger} applies, so that $e^{-tH_N}$ is positivity preserving (see Definition\-~\ref{def:positivity_preserving}). +Use Theorem\-~\ref{theo:add_inv} to show that $e^{-tH_N}$ is positivity preserving (see Definition\-~\ref{def:positivity_preserving}). Use the Perron-Frobenius theorem (Theorem\-~\ref{theo:perron_frobenius}) to conclude. \bigskip @@ -2934,24 +4407,25 @@ Prove that and use that to prove that $\psi_0\geqslant 0$. \bigskip -\problem\label{ex:feynman_hellman} (solution on p.\-~\ref{sol:feynman_hellman})\par -\smallskip -In this exercise, we will show how to compute the condensate fraction in terms of the ground state energy of an effective Hamiltonian. -Let -\begin{equation} - \tilde H_N(\epsilon):= - H_N+\epsilon\sum_{i=1}^NP_i -\end{equation} -where $P_i$ is the projector onto the constant state\-~(\ref{Pi}) (recall\-~(\ref{eta_Pi})). -Let $\tilde E_0(\epsilon)$ denote the ground state energy of $\tilde H_N(\epsilon)$. -Prove that -\begin{equation} - \eta_0= - \lim_{\displaystyle\mathop{\scriptstyle N,V\to\infty}_{\frac NV=\rho}} - \frac1N - \partial_\epsilon\tilde E_0|_{\epsilon=0} - . -\end{equation} +%\problem\label{ex:feynman_hellman} (solution on p.\-~\ref{sol:feynman_hellman})\par +%\smallskip +%In this exercise, we will show how to compute the condensate fraction in terms of the ground state energy of an effective Hamiltonian. +%Let +%\begin{equation} +% \tilde H_N(\epsilon):= +% H_N+\epsilon\sum_{i=1}^NP_i +%\end{equation} +%where $P_i$ is the projector onto the constant state\-~(\ref{Pi}) (recall\-~(\ref{eta_Pi})). +%Let $\tilde E_0(\epsilon)$ denote the ground state energy of $\tilde H_N(\epsilon)$. +%Prove that +%\begin{equation} +% \eta_0= +% \lim_{\displaystyle\mathop{\scriptstyle N,V\to\infty}_{\frac NV=\rho}} +% \frac1N +% \partial_\epsilon\tilde E_0|_{\epsilon=0} +% . +%\end{equation} + \vfill \eject @@ -3111,7 +4585,8 @@ This implies\-~(\ref{sol_softcore}). \bigskip \solution{perron_frobenius} -By Theorem\-~\ref{theo:schrodinger}, $e^{-tH_N}$ is positivity preserving. +By Lemma\-~\ref{lemma:heat_L2}, $e^{\frac tm\sum_i\Delta_i}$ is positivity preserving, and because $v\geqslant 0$, so is $e^{t\sum_{i -\end{equation} -so -\begin{equation} - \partial_\epsilon\tilde E_0|_{\epsilon=0} - = - 2\left<\partial_\epsilon\tilde\psi_0|_{\epsilon=0}\right|H_N\left|\psi_0\right> - + - \left<\psi_0\right|\partial_\epsilon\tilde H_N|_{\epsilon=0}\left|\psi_0\right> -\end{equation} -that is -\begin{equation} - \partial_\epsilon\tilde E_0|_{\epsilon=0} - = - 2E_0\left<\partial_\epsilon\tilde\psi_0|_{\epsilon=0},\psi_0\right> - + - \left<\psi_0\right|\sum_{i=1}^NP_i\left|\psi_0\right> - . -\end{equation} -In addition, -\begin{equation} - 2\left<\partial_\epsilon\tilde\psi_0|_{\epsilon=0},\psi_0\right> - = - \partial_\epsilon\left<\tilde\psi_0(\epsilon),\tilde\psi_0(\epsilon)\right>|_{\epsilon=0} - = - \partial_\epsilon1|_{\epsilon=0} - =0 - . -\end{equation} -Thus, -\begin{equation} - \frac1N - \partial_\epsilon\tilde E_0|_{\epsilon=0} - = - \frac1N\sum_{i=1}^N\left<\psi_0\right|P_i\left|\psi_0\right> - . -\end{equation} - +%\solution{feynman_hellman} +%Let $\tilde\psi_0(\epsilon)$ denote the ground state of $\tilde H_N(\epsilon)$ with $\|\tilde\psi_0(\epsilon)\|_2=1$. +%We have +%\begin{equation} +% \tilde E_0(\epsilon) +% =\left<\tilde\psi_0(\epsilon)\right|\tilde H_N(\epsilon)\left|\tilde\psi_0(\epsilon)\right> +%\end{equation} +%so +%\begin{equation} +% \partial_\epsilon\tilde E_0|_{\epsilon=0} +% = +% 2\left<\partial_\epsilon\tilde\psi_0|_{\epsilon=0}\right|H_N\left|\psi_0\right> +% + +% \left<\psi_0\right|\partial_\epsilon\tilde H_N|_{\epsilon=0}\left|\psi_0\right> +%\end{equation} +%that is +%\begin{equation} +% \partial_\epsilon\tilde E_0|_{\epsilon=0} +% = +% 2E_0\left<\partial_\epsilon\tilde\psi_0|_{\epsilon=0},\psi_0\right> +% + +% \left<\psi_0\right|\sum_{i=1}^NP_i\left|\psi_0\right> +% . +%\end{equation} +%In addition, +%\begin{equation} +% 2\left<\partial_\epsilon\tilde\psi_0|_{\epsilon=0},\psi_0\right> +% = +% \partial_\epsilon\left<\tilde\psi_0(\epsilon),\tilde\psi_0(\epsilon)\right>|_{\epsilon=0} +% = +% \partial_\epsilon1|_{\epsilon=0} +% =0 +% . +%\end{equation} +%Thus, +%\begin{equation} +% \frac1N +% \partial_\epsilon\tilde E_0|_{\epsilon=0} +% = +% \frac1N\sum_{i=1}^N\left<\psi_0\right|P_i\left|\psi_0\right> +% . +%\end{equation} \end{document} diff --git a/README b/README index dd9aef7..25c693c 100644 --- a/README +++ b/README @@ -1,4 +1,11 @@ -This directory contains the source files to typeset the article, and generate +This is a preprint of the following work: I. Jauslin, An Introduction to Lieb's +Simplified Approach, 2025, Springer. It is the version of the author's +manuscript prior to acceptance for publication and has not undergone editorial +and/or peer review on behalf of the Publisher (where applicable). The final +authenticated version is available online at: +http://dx.doi.org/10.1007/978-3-031-81393-1 + +This directory contains the source files to typeset the book, and generate the figures. This can be accomplished by running make diff --git a/bibliography/bibliography.tex b/bibliography/bibliography.tex index 2ad2ff4..b02d9cd 100644 --- a/bibliography/bibliography.tex +++ b/bibliography/bibliography.tex @@ -4,11 +4,17 @@ doi:{\tt\color{blue}\href{http://dx.doi.org/10.1126/science.269.5221.198}{10.112 \bibitem[BCS21]{BCS21}G. Basti, S. Cenatiempo, B. 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Simon - {\it Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness}, second edition, Academic Press, New York, 1975.\par\medskip + +\bibitem[RS39]{RS39}B. Rollin, F. Simon - {\it On the ``film'' phenomenon of liquid helium II}, Physica, volume\-~6, issue\-~2, pages\-~219-230, 1939,\par\penalty10000 +doi:{\tt\color{blue}\href{http://dx.doi.org/10.1016/S0031-8914(39)80013-1}{10.1016/S0031-8914(39)80013-1}}.\par\medskip + \bibitem[Ru99]{Ru99}D. Ruelle - {\it Statistical mechanics: rigorous results}, Imperial College Press, World Scientific, (first edition: Benjamin, 1969), 1999.\par\medskip \bibitem[Sc22]{Sc22}B. Schlein - {\it Bose gases in the Gross-Pitaevskii limit: A survey of some rigorous results}, The Physics and Mathematics of Elliott Lieb, The\-~90th anniversary volume II, eds R.L. Frank, A. Laptev, M. Lewin, R. Seiringer, pages\-~277-305, 2022,\par\penalty10000 @@ -69,12 +106,21 @@ doi:{\tt\color{blue}\href{http://dx.doi.org/10.4171/90-2/40}{10.4171/90-2/40}}, \bibitem[Se11]{Se11}R. Seiringer - {\it The Excitation Spectrum for Weakly Interacting Bosons}, Communications in Mathematical Physics, volume\-~306, issue\-~2, pages\-~565-578, 2011,\par\penalty10000 doi:{\tt\color{blue}\href{http://dx.doi.org/10.1007/s00220-011-1261-6}{10.1007/s00220-011-1261-6}}, arxiv:{\tt\color{blue}\href{http://arxiv.org/abs/1008.5349}{1008.5349}}.\par\medskip +\bibitem[Ta08]{Ta08}S. Tan - {\it Energetics of a strongly correlated Fermi gas}, Annals of Physics, volume\-~323, issue\-~12, pages\-~2952-2970, 2008,\par\penalty10000 +doi:{\tt\color{blue}\href{http://dx.doi.org/10.1016/j.aop.2008.03.004}{10.1016/j.aop.2008.03.004}}, arxiv:{\tt\color{blue}\href{http://arxiv.org/abs/cond-mat/0505200}{cond-mat/0505200}}.\par\medskip + +\bibitem[Ta08b]{Ta08b}S. 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Teschl - {\it Mathematical Methods in Quantum Mechanics With Applications to Schr\"odinger Operators}, Second Edition, Graduate Studies in Mathematics, volume\-~157, AMS, 2014,\par\penalty10000 {\tt\color{blue}\href{https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/schroe2.pdf}{https://www.mat.univie.ac.at/$\sim$gerald/ftp/book-schroe/schroe2.pdf}}.\par\medskip \bibitem[YY09]{YY09}H. Yau, J. Yin - {\it The Second Order Upper Bound for the Ground Energy of a Bose Gas}, Journal of Statistical Physics, volume\-~136, issue\-~3, pages\-~453-503, 2009,\par\penalty10000 doi:{\tt\color{blue}\href{http://dx.doi.org/10.1007/s10955-009-9792-3}{10.1007/s10955-009-9792-3}}, arxiv:{\tt\color{blue}\href{http://arxiv.org/abs/0903.5347}{0903.5347}}.\par\medskip -\bibitem[JaSS]{ss}I.\-~Jauslin - {\tt simplesolv} software package v0.4.1,\par\penalty10000 +\bibitem[JaSS]{ss}I.\-~Jauslin - {\tt simplesolv} software package v0.4.2,\par\penalty10000 {\tt\color{blue}\href{http://ian.jauslin.org/software/simplesolv}{http://ian.jauslin.org/software/simplesolv}}.\par\medskip diff --git a/figs/condensate.fig/2020-08-27+0.5.dat b/figs/condensate.fig/2020-08-27+0.5.dat deleted file mode 100644 index 67c0851..0000000 --- a/figs/condensate.fig/2020-08-27+0.5.dat +++ /dev/null @@ -1,8 +0,0 @@ -# potential: 0.5 e^{-|x|} -# rho E 1-eta Jastrow E - 1e-3 0.005189 0.978 0.005193 - 5e-3 0.027389 0.961 0.027395 - 1e-2 0.056223 0.959 0.05624 - 2e-2 0.11530 0.954 0.11530 - 1e-1 0.6037 0.953 0.60377 - 1e+0 6.221 0.97 6.22 diff --git a/figs/condensate.fig/2020-10-15+energy.dat b/figs/condensate.fig/2020-10-15+energy.dat new file mode 100644 index 0000000..358e2dc --- /dev/null +++ b/figs/condensate.fig/2020-10-15+energy.dat @@ -0,0 +1,15 @@ +# Potential: e^{-|x|} +# rho E E+dE eta eta+deta Jastrow E Jastrow E+dE Jastrow eta Jastrow eta+deta +1e-6 7.90e-6 7.91e-6 +1e-4 8.344e-4 8.345e-4 0.017 0.018 8.355e-4 8.356e-4 0.015 0.016 +1e-3 9.1338e-3 9.1339e-3 0.048 0.049 9.142e-3 9.143e-3 0.045 0.046 +5e-3 5.0613e-2 5.0614e-2 0.080 0.081 5.0648e-2 5.0649e-2 0.070 0.071 +1e-2 1.06107e-1 1.06108e-1 0.087 0.088 1.0627e-1 1.0628e-1 0.073 0.074 +1.5e-2 1.6337e-1 1.6338e-1 0.090 0.091 1.6342e-1 1.6343e-1 0.088 0.089 +2e-2 2.2162e-1 2.2163e-1 0.093 0.094 2.2169e-1 2.2170e-1 0.090 0.091 +5e-2 5.8044e-1 5.8045e-1 0.091 0.092 5.8046e-1 5.8047e-1 0.091 0.092 +1e-1 1.1918e+0 1.1919e+0 0.088 0.089 1.1919e+0 1.1920e+0 0.085 0.086 +1e-0 1.2430e+1 1.2431e+1 0.058 0.059 1.245e+1 1.246e+1 0.05 0.06 +1e+1 1.2544e+2 1.2545e+2 0.031 0.032 +5e+1 6.2803e+2 6.2804e+2 0.02 0.03 + diff --git a/figs/condensate.fig/Makefile b/figs/condensate.fig/Makefile index 791112d..42f4154 100644 --- a/figs/condensate.fig/Makefile +++ b/figs/condensate.fig/Makefile @@ -1,6 +1,6 @@ PROJECTNAME=condensate -SIMPLESOLV= # path/to/simplesolv +SIMPLESOLV=simplesolv DATS=simpleq.dat medeq.dat bigeq.dat PDFS=$(addsuffix .pdf, $(PROJECTNAME)) @@ -13,11 +13,11 @@ $(PDFS): $(DATS) pdflatex -jobname $(basename $@) -file-line-error $(patsubst %.pdf, %.tikz.tex, $@) simpleq.dat: - julia $(SIMPLESOLV)/main.jl -p "eq=simpleq;tolerance=1e-11;order=100;maxiter=100;v_a=0.5;minlrho=-6;nlrho=100;maxlrho=2" -M easyeq -U exp condensate_fraction_rho > $@ + julia $(SIMPLESOLV)/main.jl -p "eq=simpleq;tolerance=1e-11;order=100;maxiter=100;v_a=1;minlrho=-6;nlrho=100;maxlrho=2" -M easyeq -U exp condensate_fraction_rho > $@ medeq.dat: - julia $(SIMPLESOLV)/main.jl -p "eq=medeq;tolerance=1e-11;order=100;maxiter=100;v_a=0.5;minlrho=-6;nlrho=100;maxlrho=2" -M easyeq -U exp condensate_fraction_rho > $@ + julia $(SIMPLESOLV)/main.jl -p "eq=medeq;tolerance=1e-11;order=100;maxiter=100;v_a=1;minlrho=-6;nlrho=100;maxlrho=2" -M easyeq -U exp condensate_fraction_rho > $@ bigeq.dat: - julia -p 8 $(SIMPLESOLV)/main.jl -p "eq=bigeq;N=12;P=8;J=10;tolerance=1e-11;maxiter=21;v_a=0.5;minlrho=-6;nlrho=100;maxlrho=2" -M anyeq -U exp condensate_fraction_rho > $@ + julia -p 8 $(SIMPLESOLV)/main.jl -p "eq=bigeq;N=12;P=8;J=10;tolerance=1e-11;maxiter=21;v_a=1;minlrho=-6;nlrho=100;maxlrho=2" -M anyeq -U exp condensate_fraction_rho > $@ install: $(PDFS) diff --git a/figs/condensate.fig/bigeq.dat b/figs/condensate.fig/bigeq.dat index c8769a7..b4b1fca 100644 --- a/figs/condensate.fig/bigeq.dat +++ b/figs/condensate.fig/bigeq.dat @@ -1,100 +1,100 @@ - 1.000000000000000e-06 9.359600849904642e-04 4.769492271009877e-12 - 1.204503540258781e-06 1.026263695589573e-03 3.419474268720360e-12 - 1.450828778495940e-06 1.128447423932415e-03 1.284844855728227e-12 - 1.747528400007683e-06 1.243211901088135e-03 6.036870814265909e-14 - 2.104904144512022e-06 1.371540352505089e-03 1.465826916259999e-12 - 2.535364493970111e-06 1.514563172953764e-03 5.278911635708756e-12 - 3.053855508833412e-06 1.673162936456016e-03 6.438854837197395e-12 - 3.678379771828634e-06 1.847621600906275e-03 3.277884092656937e-12 - 4.430621457583877e-06 2.037906626012430e-03 6.922442639580641e-13 - 5.336699231206313e-06 2.244403765508714e-03 4.301594724689213e-14 - 6.428073117284319e-06 2.468275249347463e-03 3.149478739794029e-14 - 7.742636826811277e-06 2.711300000206942e-03 1.692972150200860e-14 - 9.326033468832200e-06 2.975586773983234e-03 2.485257044540253e-14 - 1.123324032978027e-05 3.263387610365739e-03 5.798705939542849e-14 - 1.353047774579808e-05 3.577021358160436e-03 1.007850792994357e-13 - 1.629750834620643e-05 3.918861287982854e-03 1.180879418675055e-13 - 1.963040650040272e-05 4.291351386383927e-03 1.164548699163096e-13 - 2.364489412645407e-05 4.697030635602372e-03 9.993404840576712e-14 - 2.848035868435805e-05 5.138553215636342e-03 7.889857713883516e-14 - 3.430469286314919e-05 5.618698153738680e-03 5.544933866100784e-14 - 4.132012400115334e-05 6.140366567754929e-03 3.505040625564625e-14 - 4.977023564332114e-05 6.706568098963386e-03 2.719353007326749e-14 - 5.994842503189409e-05 7.320399534465997e-03 2.419046911428554e-14 - 7.220809018385471e-05 7.985017865199253e-03 2.911321693643152e-14 - 8.697490026177834e-05 8.703608059981844e-03 1.665264019357716e-14 - 1.047615752789665e-04 9.479343964932427e-03 1.735486882457866e-14 - 1.261856883066021e-04 1.031533974401874e-02 1.231730471561613e-14 - 1.519911082952935e-04 1.121458911634652e-02 1.063694582767876e-14 - 1.830738280295370e-04 1.217988982497501e-02 2.605952558828706e-14 - 2.205130739903046e-04 1.321375095073367e-02 1.948459628184608e-14 - 2.656087782946686e-04 1.431828101890961e-02 2.204861946752505e-14 - 3.199267137797384e-04 1.549505586477500e-02 3.187650388653336e-14 - 3.853528593710532e-04 1.674496732600098e-02 3.041558319770699e-14 - 4.641588833612782e-04 1.806805674261242e-02 3.132688775353608e-14 - 5.590810182512228e-04 1.946333993043477e-02 3.849130191836202e-14 - 6.734150657750828e-04 2.092863160214499e-02 5.858344140792253e-14 - 8.111308307896872e-04 2.246037686139622e-02 8.179691054969686e-14 - 9.770099572992256e-04 2.405349636533224e-02 1.097457892783796e-13 - 1.176811952434999e-03 2.570125163577371e-02 1.510104805632382e-13 - 1.417474162926806e-03 2.739513888148153e-02 2.017323906157568e-13 - 1.707352647470692e-03 2.912482328833103e-02 2.635554179184661e-13 - 2.056512308348653e-03 3.087812926296352e-02 3.398621654197443e-13 - 2.477076355991711e-03 3.264110309962052e-02 4.285413087823841e-13 - 2.983647240283340e-03 3.439816133650226e-02 5.302517700005698e-13 - 3.593813663804629e-03 3.613233114795761e-02 6.403992903217449e-13 - 4.328761281083062e-03 3.782558079607085e-02 7.571924192093096e-13 - 5.214008287999690e-03 3.945923077971335e-02 8.735665611670509e-13 - 6.280291441834260e-03 4.101443056577509e-02 9.852419079005244e-13 - 7.564633275546291e-03 4.247268081094784e-02 1.086591967833584e-12 - 9.111627561154896e-03 4.381637597402372e-02 1.171028110980761e-12 - 1.097498765493057e-02 4.502933773539811e-02 1.235182529056386e-12 - 1.321941148466031e-02 4.609730735888347e-02 1.273413495244793e-12 - 1.592282793341094e-02 4.700836649313180e-02 1.288870490747045e-12 - 1.917910261672489e-02 4.775326110412755e-02 1.277774794460160e-12 - 2.310129700083163e-02 4.832561126937697e-02 1.245436832753992e-12 - 2.782559402207126e-02 4.872199901018250e-02 1.194094596784815e-12 - 3.351602650938848e-02 4.894193587141005e-02 1.129149381775895e-12 - 4.037017258596558e-02 4.898772053354203e-02 1.056418952921306e-12 - 4.862601580065354e-02 4.886420356210291e-02 9.797825696635614e-13 - 5.857020818056673e-02 4.857848093382484e-02 9.036039641219436e-13 - 7.054802310718646e-02 4.813954002928141e-02 8.309184154224132e-13 - 8.497534359086456e-02 4.755788150120432e-02 7.632736986718884e-13 - 1.023531021899027e-01 4.684513825031006e-02 7.009835850851132e-13 - 1.232846739442068e-01 4.601370925687214e-02 6.445926849679039e-13 - 1.484968262254467e-01 4.507642184667945e-02 5.938594915648704e-13 - 1.788649529057435e-01 4.404623166683212e-02 5.471692016966516e-13 - 2.154434690031887e-01 4.293596563723694e-02 5.037037371246892e-13 - 2.595024211399737e-01 4.175810970901533e-02 4.628803528432183e-13 - 3.125715849688241e-01 4.052464054410850e-02 4.243706234593440e-13 - 3.764935806792471e-01 3.924689825998715e-02 3.870128277902043e-13 - 4.534878508128591e-01 3.793549610670589e-02 3.506993848968064e-13 - 5.462277217684348e-01 3.660026225836260e-02 3.158617612429666e-13 - 6.579332246575682e-01 3.525020868323555e-02 2.820376102788483e-13 - 7.924828983539185e-01 3.389352218106337e-02 2.495811928366868e-13 - 9.545484566618347e-01 3.253757302873513e-02 2.187355418163057e-13 - 1.149756995397738e+00 3.118893716272646e-02 1.896018293540065e-13 - 1.384886371393875e+00 2.985342837493234e-02 1.627002158331657e-13 - 1.668100537200059e+00 2.853613755533530e-02 1.381782671776070e-13 - 2.009233002565050e+00 2.724147654488262e-02 1.162433870344956e-13 - 2.420128264794383e+00 2.597322464354741e-02 9.667411489592391e-14 - 2.915053062825182e+00 2.473457624087447e-02 7.921718375131266e-14 - 3.511191734215135e+00 2.352818839610205e-02 6.467146452368413e-14 - 4.229242874389508e+00 2.235622749416585e-02 5.197716963886527e-14 - 5.094138014816386e+00 2.122041434752589e-02 4.164284351060715e-14 - 6.135907273413176e+00 2.012206730828386e-02 3.267501010541636e-14 - 7.390722033525790e+00 1.906214310748888e-02 2.631758984864249e-14 - 8.902150854450392e+00 1.804127525584338e-02 1.984138301410465e-14 - 1.072267222010325e+01 1.705980992860735e-02 1.572328524590868e-14 - 1.291549665014885e+01 1.611783932288287e-02 1.184638961517491e-14 - 1.555676143930475e+01 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2.718588242732946e+01 1.895617187491327e-02 5.927252259704543e-14 + 3.274549162877732e+01 1.779899778957120e-02 4.383104277609757e-14 + 3.944206059437664e+01 1.670297232162228e-02 3.275121851070678e-14 + 4.750810162102803e+01 1.566577468297768e-02 2.438653727506966e-14 + 5.722367659350220e+01 1.468507755645780e-02 1.781857842665750e-14 + 6.892612104349709e+01 1.375856030036783e-02 1.398489991139448e-14 + 8.302175681319753e+01 1.288392037060158e-02 1.067384461164944e-14 + 1.000000000000000e+02 1.205888314637288e-02 8.541168489759710e-15 diff --git a/figs/condensate.fig/condensate.gnuplot b/figs/condensate.fig/condensate.gnuplot index ec27372..0155b4c 100644 --- a/figs/condensate.fig/condensate.gnuplot +++ b/figs/condensate.fig/condensate.gnuplot @@ -8,14 +8,14 @@ unset mxtics #set ytics 0.6, 0.1 set mytics 2 -set xrange [4e-8:100] +set xrange [1e-8:100] -set yrange [:0.06] +set yrange [0.9:1] # default output canvas size: 12.5cm x 8.75cm set term lua tikz size 8,6 standalone -set key top left box linetype rgbcolor"#999999" width 0.3 height 0.3 spacing 1.3 +set key bottom left box linetype rgbcolor"#999999" width 0.3 height 0.3 spacing 1.3 # set linestyle @@ -30,12 +30,13 @@ set pointsize 1 set logscale x -a=0.7666858699084325 +#a=0.7666858699084325 +a=1.2543564105910647 plot \ - "simpleq.dat" using 1:($3<1e-5 ? $2 : 1/0) with lines ls 1 title "simple",\ - "medeq.dat" using 1:($3<1e-5 ? $2 : 1/0) with lines ls 3 title "med",\ - "bigeq.dat" using 1:($3<1e-5 ? $2 : 1/0) with lines ls 5 title "big",\ - "2020-08-27+0.5.dat" using 1:(1-$3) with points ls 2 title "QMC" ,\ - 8./(3*sqrt(pi))*sqrt(x*a**3) ls 6 title "Bog" + "simpleq.dat" using 1:($3<1e-5 ? 1-$2 : 1/0) with lines ls 1 title "simple",\ + "medeq.dat" using 1:($3<1e-5 ? 1-$2 : 1/0) with lines ls 3 title "med",\ + "bigeq.dat" using 1:($3<1e-5 ? 1-$2 : 1/0) with lines ls 5 title "big",\ + "2020-10-15+energy.dat" using 1:(1-$4) with points ls 2 title "QMC" ,\ + 1-8./(3*sqrt(pi))*sqrt(x*a**3) ls 6 title "Bog" diff --git a/figs/condensate.fig/medeq.dat b/figs/condensate.fig/medeq.dat index dd9f619..01cb81a 100644 --- a/figs/condensate.fig/medeq.dat +++ b/figs/condensate.fig/medeq.dat @@ -1,100 +1,100 @@ - 1.000000000000000e-06 9.996791129944504e-04 2.557931963013874e-14 - 1.204503540258781e-06 1.096049436651035e-03 1.355586210122227e-14 - 1.450828778495940e-06 1.201592925494163e-03 1.225664725981035e-14 - 1.747528400007683e-06 1.317158886205431e-03 1.598257741770033e-14 - 2.104904144512022e-06 1.443670377419786e-03 1.553247893694121e-14 - 2.535364493970111e-06 1.582129599439783e-03 2.230882620498033e-14 - 3.053855508833412e-06 1.733623466010337e-03 1.130567634193424e-14 - 3.678379771828634e-06 1.899329292079710e-03 9.168136807150408e-15 - 4.430621457583877e-06 2.080520546462278e-03 8.941527845815565e-15 - 5.336699231206313e-06 2.278572579850316e-03 1.057140657470087e-14 - 6.428073117284319e-06 2.494968213624501e-03 1.682809007854741e-14 - 7.742636826811277e-06 2.731303057010942e-03 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+PDFS=$(addsuffix .pdf, $(PROJECTNAME)) +TEXS=$(addsuffix .tikz.tex, $(PROJECTNAME)) + +all: $(PDFS) + +$(PDFS): $(DATS) + gnuplot $(patsubst %.pdf, %.gnuplot, $@) > $(patsubst %.pdf, %.tikz.tex, $@) + pdflatex -jobname $(basename $@) -file-line-error $(patsubst %.pdf, %.tikz.tex, $@) + +simpleq0001.dat: + julia $(SIMPLESOLV)/main.jl -p "rho=1e-4;minlrho_init=-6;nlrho_init=20;v_a=16;N=12;P=8;J=10;tolerance=1e-11;maxiter=21;xmin=0;nx=100;xmax=55" -M simpleq-Kv -U exp 2pt > $@ +medeq0001.dat: + julia $(SIMPLESOLV)/main.jl -p "rho=1e-4;minlrho_init=-6;nlrho_init=20;v_a=16;eq=medeq;N=12;P=8;J=10;tolerance=1e-11;maxiter=21;xmin=0;nx=100;xmax=55" -M anyeq -U exp 2pt > $@ +bigeq0001.dat: + julia $(SIMPLESOLV)/main.jl -p "rho=1e-4;minlrho_init=-6;nlrho_init=20;v_a=16;eq=bigeq;N=12;P=8;J=10;tolerance=1e-11;maxiter=21;xmin=0;nx=100;xmax=55" -M anyeq -U exp 2pt > $@ + +simpleq02.dat: + julia $(SIMPLESOLV)/main.jl -p "rho=2e-2;minlrho_init=-6;nlrho_init=60;v_a=16;N=12;P=8;J=10;tolerance=1e-11;maxiter=21;xmin=0;nx=100;xmax=10" -M simpleq-Kv -U exp 2pt > $@ +medeq02.dat: + julia $(SIMPLESOLV)/main.jl -p "rho=2e-2;minlrho_init=-6;nlrho_init=60;v_a=16;eq=medeq;N=12;P=8;J=10;tolerance=1e-11;maxiter=21;xmin=0;nx=100;xmax=10" -M anyeq -U exp 2pt > $@ +bigeq02.dat: + julia $(SIMPLESOLV)/main.jl -p "rho=2e-2;minlrho_init=-6;nlrho_init=60;v_a=16;eq=bigeq;N=12;P=8;J=10;tolerance=1e-11;maxiter=21;xmin=0;nx=100;xmax=10" -M anyeq -U exp 2pt > $@ + +install: $(PDFS) + cp $^ $(INSTALLDIR)/ + +clean-aux: + rm -f $(addsuffix .tikz.tex, $(PROJECTNAME)) + rm -f $(addsuffix .aux, $(PROJECTNAME)) + rm -f $(addsuffix .log, $(PROJECTNAME)) + +clean-dat: + rm -f $(DATS) + +clean-tex: + rm -f $(PDFS) + +clean: clean-aux clean-tex diff --git a/figs/correlation.fig/bigeq0001.dat b/figs/correlation.fig/bigeq0001.dat new file mode 100644 index 0000000..2f6ea5f --- /dev/null +++ b/figs/correlation.fig/bigeq0001.dat @@ -0,0 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0.3 spacing 1.3 + + +# set linestyle +set style line 1 linetype rgbcolor "#4169E1" linewidth 2 dashtype "." +set style line 2 linetype rgbcolor "#DC143C" linewidth 1 +set style line 3 linetype rgbcolor "#32CD32" linewidth 2 dashtype "-" +set style line 4 linetype rgbcolor "#4B0082" linewidth 2 +set style line 5 linetype rgbcolor "#DAA520" linewidth 2 +set style line 6 linetype rgbcolor "#555500" linewidth 2 + +set pointsize 0.5 + +rho=1e-4 + +plot \ + "simpleq0001.dat" using 1:($2/rho**2) with lines ls 1 title "simple",\ + "medeq0001.dat" using 1:($2/rho**2) with lines ls 3 title "med",\ + "bigeq0001.dat" using 1:($2/rho**2) with lines ls 5 title "big",\ + "2020-10-01+correlation16-0001.dat" using 1:2 ls 2 title "QMC" + diff --git a/figs/correlation.fig/correlation02.gnuplot b/figs/correlation.fig/correlation02.gnuplot new file mode 100644 index 0000000..1637b98 --- /dev/null +++ b/figs/correlation.fig/correlation02.gnuplot @@ -0,0 +1,28 @@ +set title "$\\rho=0.02$" +set ylabel "$\\frac{C_2}{\\rho^2}$" norotate offset -1,0 +set xlabel "$|x|$" + +# default output canvas size: 12.5cm x 8.75cm +set term lua tikz size 8,6 standalone + +set key bottom right box linetype rgbcolor"#999999" width 0.3 height 0.3 spacing 1.3 + + +# set linestyle +set style line 1 linetype rgbcolor "#4169E1" linewidth 2 dashtype "." +set style line 2 linetype rgbcolor "#DC143C" linewidth 1 +set style line 3 linetype rgbcolor "#32CD32" linewidth 2 dashtype "-" +set style line 4 linetype rgbcolor "#4B0082" linewidth 2 +set style line 5 linetype rgbcolor "#DAA520" linewidth 2 +set style line 6 linetype rgbcolor "#555500" linewidth 2 + +set pointsize 0.5 + +rho=2e-2 + +plot \ + "simpleq02.dat" using 1:($2/rho**2) with lines ls 1 title "simple",\ + "medeq02.dat" using 1:($2/rho**2) with lines ls 3 title "med",\ + "bigeq02.dat" using 1:($2/rho**2) with lines ls 5 title "big",\ + "2020-08-27+correlation.dat" using 1:2 ls 2 title "QMC" + diff --git a/figs/correlation.fig/medeq0001.dat 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1.592282793341094e-02 1.814245009403908e-01 2.339502787826644e-12 1.917910261672489e-02 2.209107486051246e-01 1.688210571908239e-12 2.310129700083163e-02 2.687825259198435e-01 1.167659413279469e-12 - 2.782559402207126e-02 3.267664771683836e-01 7.775556337126751e-13 + 2.782559402207126e-02 3.267664771683836e-01 7.775556337127040e-13 3.351602650938848e-02 3.969379725867567e-01 4.949300609691499e-13 4.037017258596558e-02 4.817921803787671e-01 3.047835972157860e-13 - 4.862601580065354e-02 5.843299025667947e-01 1.744034034347572e-13 + 4.862601580065353e-02 5.843299025667946e-01 1.744349956560391e-13 5.857020818056673e-02 7.081611855740870e-01 2.145108372619088e-13 7.054802310718646e-02 8.576303130499369e-01 1.004811457230378e-13 - 8.497534359086456e-02 1.037966511958588e+00 4.718419633908083e-14 - 1.023531021899027e-01 1.255465580586508e+00 2.639823583877913e-14 + 8.497534359086456e-02 1.037966511958588e+00 5.257246692725826e-14 + 1.023531021899027e-01 1.255465580586508e+00 2.633530652792523e-14 1.232846739442068e-01 1.517708710502109e+00 1.889177952517916e-13 - 1.484968262254467e-01 1.833826060755470e+00 1.262402568827356e-13 + 1.484968262254467e-01 1.833826060755470e+00 1.262402586202156e-13 1.788649529057435e-01 2.214814195859386e+00 2.307786324072868e-13 - 2.154434690031887e-01 2.673918372281924e+00 3.831494558709393e-14 + 2.154434690031887e-01 2.673918372281924e+00 3.831493630360338e-14 2.595024211399737e-01 3.227092915163674e+00 6.546666422710970e-14 3.125715849688241e-01 3.893555644968086e+00 8.786410093299553e-14 - 3.764935806792471e-01 4.696455586182800e+00 3.094041097523017e-14 + 3.764935806792472e-01 4.696455586182801e+00 3.094040729325163e-14 4.534878508128591e-01 5.663677129915349e+00 1.346721396138190e-14 - 5.462277217684348e-01 6.828808565598630e+00 2.348290345391161e-14 + 5.462277217684348e-01 6.828808565598630e+00 2.348301972394285e-14 6.579332246575682e-01 8.232308608511897e+00 6.075413859670748e-14 - 7.924828983539185e-01 9.922911427699711e+00 3.551771518542288e-14 - 9.545484566618347e-01 1.195931896203401e+01 8.585404990085891e-14 + 7.924828983539186e-01 9.922911427699711e+00 3.551774064603687e-14 + 9.545484566618347e-01 1.195931896203401e+01 8.585405122642469e-14 1.149756995397738e+00 1.441223928837902e+01 4.176255867225544e-14 1.384886371393875e+00 1.736684182166722e+01 1.114486677072019e-13 - 1.668100537200059e+00 2.092571459958015e+01 5.653897977104765e-14 + 1.668100537200059e+00 2.092571459958015e+01 5.653898277771886e-14 2.009233002565050e+00 2.521242633713612e+01 1.048273767790111e-14 2.420128264794383e+00 3.037581693424489e+01 8.488597666410238e-15 2.915053062825182e+00 3.659516541136894e+01 5.400259352206385e-14 @@ -94,7 +94,7 @@ 3.274549162877732e+01 4.114552381359539e+02 1.315017988355486e-14 3.944206059437664e+01 4.956068009543682e+02 1.004394652722235e-14 4.750810162102803e+01 5.969676580241880e+02 5.364915392605858e-14 - 5.722367659350220e+01 7.190571706518009e+02 5.378285825138610e-14 + 5.722367659350220e+01 7.190571706518009e+02 5.378285824908168e-14 6.892612104349709e+01 8.661144220395618e+02 5.426311368936924e-14 - 8.302175681319753e+01 1.043245402953268e+03 9.067219514460403e-12 - 1.000000000000000e+02 1.256600297381089e+03 6.252162647544124e-12 + 8.302175681319753e+01 1.043245402953268e+03 9.067219782243920e-12 + 1.000000000000000e+02 1.256600297381089e+03 6.252158470994988e-12 diff --git a/libs/ian.cls b/libs/ian.cls index 5f4a336..8d7a4c8 100644 --- a/libs/ian.cls +++ b/libs/ian.cls @@ -26,9 +26,11 @@ \DeclareOption{no_subsection_in_eq}{\subsectionsineqfalse} \DeclareOption{no_subsection_in_fig}{\subsectionsinfigfalse} \DeclareOption{no_subsection_in_all}{\subsectionsineqfalse\subsectionsinfigfalse} +\DeclareOption{a4paper}{\afourpaper} +\DeclareOption{letterpaper}{\letterpaper} \def\ian@defaultoptions{ - \ExecuteOptions{section_in_all, no_subsection_in_all} + \ExecuteOptions{section_in_all, no_subsection_in_all, a4paper} \ProcessOptions %% required packages @@ -41,9 +43,56 @@ \pagestyle{plain} } -%% paper dimensions -\setlength\paperheight{297mm} -\setlength\paperwidth{210mm} +%% paper sizes +\def\afourpaper{ + %% paper dimensions + \setlength\paperheight{297mm} + \setlength\paperwidth{210mm} + %% offsets + \hoffset=-1in + \voffset=-1in + %% horizontal margins + \oddsidemargin=25mm + \evensidemargin=25mm + %% vertical margin + \topmargin=15mm + %% body size + \textwidth=160mm + \textheight=247mm + %% header size and margin + \headheight=5mm + \headsep=5mm + %% footer size + %\footskip=30pt + %% margin size and margin + \marginparwidth=15mm + \marginparsep=5mm +} + +\def\letterpaper{ + \setlength\paperheight{11in} + \setlength\paperwidth{8.5in} + %% offsets + \hoffset=-1in + \voffset=-1in + %% horizontal margins + \oddsidemargin=1in + \evensidemargin=1in + %% vertical margin + \topmargin=0.5in + %% body size + \textwidth=6.5in + \textheight=9in + %% header size and margin + \headheight=0.25in + \headsep=0.25in + %% footer size + %\footskip=30pt + %% margin size and margin + \marginparwidth=0.4in + \marginparsep=0.2in +} + %% fonts \input{size11.clo} @@ -53,12 +102,6 @@ \DeclareOldFontCommand{\bf}{\normalfont\bfseries}{\mathbf} \DeclareOldFontCommand{\it}{\normalfont\itshape}{\mathit} -%% text dimensions -\hoffset=-50pt -\voffset=-72pt -\textwidth=460pt -\textheight=704pt - %% remove default indentation \parindent=0pt @@ -120,9 +163,9 @@ \hyperlink{ln.\csname label@##1@hl\endcsname}{{\color{blue}\safe\csname label@##1\endcsname}}% \else% \ifcsname label@##1\endcsname% - {\color{blue}\csname ##1\endcsname}% - \else% - {\bf ??}% + {\color{blue}\csname ##1\endcsname}% + \else% + {\bf ??}% \fi% \fi% } @@ -136,9 +179,12 @@ \csname ##1\endcsname% \else% {\bf ??}% - \fi% - \fi% - } + {\color{blue}\csname ##1\endcsname}% + \else% + {\bf ??}% + \fi% + \fi% + } } @@ -435,12 +481,13 @@ % booleans (write section or subsection in equation number) \newif\ifsectionsinfig \newif\ifsubsectionsinfig -% width of figures -\newlength\figwidth -\setlength\figwidth\textwidth -\addtolength\figwidth{-2.5cm} % caption \def\defcaption{ + % width of figures + \newlength\figwidth + \setlength\figwidth\textwidth + \addtolength\figwidth{-2.5cm} + \long\def\caption##1{% \stepcounter{figcount}% % @@ -530,34 +577,6 @@ %\def\seqskip{\vfill\pagebreak} } -%% start exercises -\def\exercises{ - \vfill - \pagebreak - - % counter - \setcounter{sectioncount}0 - \setcounter{seqcount}0 - - % prefix - \def\sectionprefix{E} - - \hrefanchor - - % write - {\bf \LARGE Exercises}\par\penalty10000\bigskip\penalty10000 - - % add a mention in the table of contents - \iftoc - % save lncount in aux variable which is written to toc - \immediate\write\tocoutput{\noexpand\expandafter\noexpand\edef\noexpand\csname toc@exercises\endcsname{\thelncount}} - \write\tocoutput{\noexpand\tocexercises{\thepage}}\penalty10000 - \fi - - \def\eqprefix{E} - \sectionsineqfalse -} - %% bibliography % size of header @@ -593,7 +612,7 @@ %% cite command % @tempswa is set to true when there is an optional argument -\newif\@tempswa +\newif\if@tempswa \def\cite{% % check whether there is an optional argument (if there is none, add on empty '[]') \@ifnextchar [{\@tempswatrue\@citex}{\@tempswafalse\@citex[]}% @@ -692,13 +711,6 @@ {\hyperlink{ln.\csname toc@references\endcsname}{{\color{blue}\bf References}\leaderfill#1}}\par \smallskip } -\def\tocexercises#1{ - \medskip - \setcounter{tocsectioncount}0 - {\hyperlink{ln.\csname toc@exercises\endcsname}{{\color{blue}\bf Exercises}\leaderfill#1}}\par - \smallskip - \def\tocsectionprefix{E} -} %% definitions that must be loaded at begin document @@ -710,6 +722,43 @@ \defcaption } +%% start exercises +\def\exercises{ + \vfill + \pagebreak + + % counter + \setcounter{sectioncount}0 + \setcounter{seqcount}0 + + % prefix + \def\sectionprefix{E} + + \hrefanchor + + % write + {\bf \LARGE Exercises}\par\penalty10000\bigskip\penalty10000 + + % add a mention in the table of contents + \iftoc + % save lncount in aux variable which is written to toc + \immediate\write\tocoutput{\noexpand\expandafter\noexpand\edef\noexpand\csname toc@exercises\endcsname{\thelncount}} + \write\tocoutput{\noexpand\tocexercises{\thepage}}\penalty10000 + \fi + + \def\eqprefix{E} + \sectionsineqfalse +} + + +\def\tocexercises#1{ + \medskip + \setcounter{tocsectioncount}0 + {\hyperlink{ln.\csname toc@exercises\endcsname}{{\color{blue}\bf Exercises}\leaderfill#1}}\par + \smallskip + \def\tocsectionprefix{E} +} + %% end \ian@defaultoptions \endinput diff --git a/libs/problemset.sty b/libs/problemset.sty index 9d14474..a385294 100644 --- a/libs/problemset.sty +++ b/libs/problemset.sty @@ -108,8 +108,8 @@ \def\questionformat#1{{\bf(#1)} } \def\solution#1{ - \makelink{sol:#1}{\thepage} {\bf Solution\-~\ref{ex:#1}}\par\penalty10000\smallskip\penalty10000 + \makelink{sol:#1}{\thepage} \def\eqprefix{S\textref{ex:#1}.} \resetpointcounter \setcounter{seqcount}{0}