Update to v0.1:
Added: references to [Nelson, 1964] and [Reed, Simon, 1975]. Fixed: sign of Laplacian in definition of \mathfrak K_e. Added: references to proofs in introduction. Fixed: miscellaneous typos.
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@ -114,7 +114,7 @@ Along with the computation of the low density energy of the simple equation in o
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,\quad
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\frac{2e}{\rho}=\int (1-u(x))v(x)\ dx
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\end{equation}
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to be solved for an integrable function $u$ on $\R^3$ where $v$ is a given non-negative radial function representing a repulsive interaction between particles with $(1+|x|^4)v\in L^1(\R^3)\cap L^2(\R^3)$, and where
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to be solved for an integrable function $u$ on $\R^3$ where $v$ is a given non-negative radial function representing a repulsive interaction between particles with $(1+|x|^4)v\in L^1(\R^3)\cap L^2(\R^3)$, and
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where $e$ and $\rho$ are positive parameters representing, respectively, the energy per-particle and the density in the ground state of a Bose gas, and are related by the second equation in\-~(\ref{simp}). As we explain below, the solution $u(x)$ specifies a pair correlation function for the Bose gas in terms of which many observable of physical interest can be computed.
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This system was first introduced in \cite{Li63,LS64,LL64} and the equation on the left is referred to here as the {\em simple equation}; it results from applying some approximations to a more complicated equation derived in \cite{Li63}.
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For the reader interested in the origins of this equation, we give a brief account of its derivation and motivation.
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@ -123,15 +123,15 @@ Along with the computation of the low density energy of the simple equation in o
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H_N:=-\frac12\sum_{i=1}^N\Delta_i+\sum_{i<j}v(x_i-x_j)
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\end{equation}
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for $N$ particles in a cubic box of finite volume $V$ with periodic boundary conditions.
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The ground state eigenfunction $\psi_0$ is unique and non-negative, as can be shown using the Perron-Frobenius theorem, and thus we may normalize $\psi_0$ to obtain a probability measure. This is not the usual probability measure associated a quantum state, which would be quadratic in the wave function, but since $\psi_0$ is non-negative and integrable ($\|\psi_0\|_1 \leqslant V^{1/2}\|\psi_0\|_2$), we may use it directly to define a probability measure, and this is the starting point of \cite{Li63}. Because particles interact pairwise, the ground state energy and other observables can be calculated in terms of the two-point correlation function associated to this probability measure.
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In \cite{Li63}, it is argued that, under a few physically motivated approximations, in the thermodynamic limit, in which the number of particles $N$ and the volume of the gas $V$ are taken to infinity, with $\rho:=\frac NV$ fixed, the limiting two-point correlation function
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\begin{equation}
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g(x_1-x_2) := \lim_{N,V\to\infty, N/V = \rho}\frac{V^2\int dx_3\cdots dx_N\ \psi_0(x_1,x_2,x_3,\dots,x_N)}{\int dy_1\cdots dy_N\ \psi_0(y_1,\dots,y_N )}
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\end{equation}
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The function $u(x)$ in \eqref{simp} is then defined as $u(x):=1- g(x)$. Note that since by definition $g(x) \geqslant 0$, $u(x) \leqslant 1$.
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The ground state eigenfunction $\psi_0$ is unique and non-negative, as can be shown using the Perron-Frobenius theorem, and thus we may normalize $\psi_0$ to obtain a probability measure. This is not the usual probability measure associated to a quantum state, which would be quadratic in the wave function, but since $\psi_0$ is non-negative and integrable ($\|\psi_0\|_1 \leqslant V^{1/2}\|\psi_0\|_2$), we may use it directly to define a probability measure, and this is the starting point of \cite{Li63}. Because particles interact pairwise, the ground state energy and other observables can be calculated in terms of the two-point correlation function associated to this probability measure:
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\begin{equation}
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g_N(x_1-x_2) := \lim_{N,V\to\infty, N/V = \rho}\frac{V^2\int dx_3\cdots dx_N\ \psi_0(x_1,x_2,x_3,\dots,x_N)}{\int dy_1\cdots dy_N\ \psi_0(y_1,\dots,y_N )}
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\end{equation}
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In \cite{Li63}, under a few physically motivated approximations, in the thermodynamic limit, in which the number of particles $N$ and the volume of the gas $V$ are taken to infinity, with $\rho:=\frac NV$ fixed, an equation for the limiting two-point correlation function $g_{\infty}$ is derived.
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The function $u(x)$ in \eqref{simp} is then defined as $u(x):=1- g_{\infty}(x)$. Note that since by definition $g_{\infty}(x) \geqslant 0$, $u(x) \leqslant 1$.
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Because the ground state expected values of many physical observables can be calculated in terms of $g$, any method for computing $g$ that bypasses direct solution of the $N$-body Schr\"odinger equation for the Hamiltonian \eqref{ham0} provides an effective means for the computation of these values, and this motivates the study of the simple equation system\-~(\ref{simp}). Indeed, the ground state energy per particle is given in terms of $g$ by the second equation in \eqref{simp}. There is so far no rigorous derivation of \eqref{simp} from the $N$-body Schr\"odinger equation, and hence there is no mathematical
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Because the expected values in the ground state of many physical observables can be calculated in terms of $g$, any method for computing $g$ that bypasses directly solving the $N$-body Schr\"odinger equation for the Hamiltonian \eqref{ham0} provides an effective means for the computation of these values, and this motivates the study of the simple equation system\-~(\ref{simp}). Indeed, the ground state energy per particle is given in terms of $g$ by the second equation in \eqref{simp}. There is so far no rigorous derivation of \eqref{simp} from the $N$-body Schr\"odinger equation, and hence there is no mathematical
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understanding of how closely the solutions of\-~(\ref{simp}) approximate the actual two point correlation function associated to the $N$-body ground state $\psi_0$. However, we have conducted extensive numerical work, about\-~(\ref{simp}) and other, more refined equations, and have found that these equations are surprisingly accurate.
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Details on the numerical results will be published elsewhere\-~\cite{CHe}.
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@ -150,7 +150,7 @@ Along with the computation of the low density energy of the simple equation in o
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If it is indeed true that the simple equation describes the many-body Bose gas in the thermodynamic limit with meaningful accuracy, then it seems important to understand this equation beyond simple numerics.
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We have started this effort in a previous publication\-~\cite{CJL20}, where we showed that, under the assumption that $v\geqslant 0$ and that $v\in L_1(\mathbb R^d)\cap L_{\frac32+\epsilon}(\mathbb R^d)$ but not necessarily radial, then in each dimension $d$, for each $e>0$, there is a unique value $\rho(e)$ for which \eqref{simp} has an integrable solution satisfying $u \leqslant 1$, and for each $e>0$, there is exactly one integrable solution $u$ with $u \leqslant 1$. (Recall that $u\leqslant 1$ is equivalent to $g\geqslant 0$, a necessary condition for the solution to be physically meaningful.) We also proved that all such solutions are necessarily positive, so that
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We have started this effort in a previous publication\-~\cite{CJL20}, where we showed that, under the assumption that $v\geqslant 0$ and that $v\in L^1(\mathbb R^d)\cap L^{\frac32+\epsilon}(\mathbb R^d)$ but not necessarily radial, then in each dimension $d$, for each $e>0$, there is a unique value $\rho(e)$ for which \eqref{simp} has an integrable solution satisfying $u \leqslant 1$, and for each $e>0$, there is exactly one integrable solution $u$ with $u \leqslant 1$. (Recall that $u\leqslant 1$ is equivalent to $g\geqslant 0$, a necessary condition for the solution to be physically meaningful.) We also proved that all such solutions are necessarily non-negative, so that
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\begin{equation}\label{uinf}
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0\leqslant u(x)\leqslant 1
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.
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@ -165,7 +165,7 @@ Along with the computation of the low density energy of the simple equation in o
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In addition, we showed that, under the further assumption that $v$ is of positive type (its Fourier transform is non-negative), the quantity $e$ defined in\-~(\ref{simp}) coincides with the ground state-energy per particle of the many-body Bose gas, asymptotically both for low and high values of $\rho$.
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In addition, we showed that, under the further assumption that $v$ is of positive type (its Fourier transform is non-negative), the quantity $e$ defined in\-~(\ref{simp}) coincides with the ground state-energy per particle of the many-body Bose gas, asymptotically both for small and large values of $\rho$.
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Finally, we showed that, if the potential $v$ is spherically symmetric and decays exponentially, then $u\sim|x|^{-4}$ for large $|x|$.
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In the present paper we take this analysis further, and prove some of the conjectures in\-~\cite{CJL20}, namely that the map $\rho\mapsto e(\rho)$ is strictly monotone increasing for small and for large $\rho$, as well as the fact that the map $\rho\mapsto \rho e(\rho)$ is convex for small values of $\rho$.
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@ -188,13 +188,13 @@ Along with the computation of the low density energy of the simple equation in o
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While the results presented in this paper may seem disparate, for the most part they are obtained through the use of a common set of mathematical tools.
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To see this, let us first consider the monotonicity result.
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To prove that the map $e\mapsto\rho(e)$ is monotone increasing,
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formally differentiate\-~(\ref{simp}) with respect to $e$, and find that, denoting derivatives with respect to $e$ by primes, to find
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formally differentiate\-~(\ref{simp}) with respect to $e$, and find that, denoting derivatives with respect to $e$ by primes,
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\begin{equation}\label{upform}
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u'=\mathfrak K_e(-4u+2\rho u\ast u+2\rho'u\ast u)
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\end{equation}
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with
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\begin{equation}\label{bg6}
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\mathfrak{K}_e = (\Delta + v + 4e(1 - C_{\rho u}))^{-1}
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\mathfrak{K}_e = (-\Delta + v + 4e(1 - C_{\rho u}))^{-1}
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\end{equation}
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in which $C_{\rho u}$ denotes the convolution by $\rho u$. Now, differentiating the second equation in \eqref{simp} in $e$ yields
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\begin{equation}\label{bg6X}
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@ -207,7 +207,7 @@ Along with the computation of the low density energy of the simple equation in o
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Justifying these formal calculations and analyzing the resulting expression for $\rho'$, we can prove its strict positivity at all sufficiently low or high densities, and in some cases, depending on $v$, for all densities. It is easy to see that the same operator $\mathfrak{K}_e$ will again show up in the computations we do to prove convexity of $e\rho(e)$. It is probably less clear that it will again show up when we derive formulas for other observable such as the condensate fraction, and we now explain why this is the case.
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Justifying these formal calculations and analyzing the resulting expression for $\rho'$, we will prove its strict positivity at all sufficiently low or high densities, and in some cases, depending on $v$, for all densities (see Theorem~\ref{Mon}). It is easy to see that the same operator $\mathfrak{K}_e$ will again show up in the computations we do to prove convexity of $e\rho(e)$. It is probably less clear that it will again show up when we derive formulas for other observable such as the condensate fraction, and we now explain why this is the case.
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Let $A$ be a self adjoint operator on the $N$-particle Hilbert space, representing some observable whose ground state expectation value $\langle \psi_0,A\psi_0\rangle$ we would like to evaluate.
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@ -250,9 +250,9 @@ Along with the computation of the low density energy of the simple equation in o
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.
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\label{eta}
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\end{equation}
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Differentiating \eqref{simpleq_eta} leads once more to the operator $\mathfrak{K}_\mu$.
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Note that, since approximations were made in computing the two particle correlation function, it is not immediately clear that the quantity $\eta$ defined in \eqref{etaf} satisfies $0 \leqslant \eta \leqslant 1$.
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In the rest of this paper, we always use $\eta$ to mean the quantity defined in \eqref{eta}, and not the true condensate fraction, defined in \eqref{cond2}.
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Differentiating \eqref{simpleq_eta} leads once more to the operator $\mathfrak{K}_{e_\mu}$.
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Note that, since approximations were made in computing the two-point correlation function, it is not immediately clear that the quantity $\eta$ defined in \eqref{etaf} satisfies $0 \leqslant \eta \leqslant 1$.
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In the rest of this paper, we always use $\eta$ to mean the quantity defined in \eqref{eta}, and not the true uncondensed fraction, defined in \eqref{cond2}.
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We shall see that at least for small $\rho$, the approximation is very good.
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\indent
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@ -272,7 +272,7 @@ A well known prediction\-~\cite{CAL09} is that, for a delta-function potential,
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\label{tan}
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\end{equation}
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which is knwon as the {\it universal Tan relation}\-~\cite{Ta08,Ta08b,Ta08c}.
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We have found that the simple equation reproduces this prediction, even when the potential is finite, when the density is asymptotically small, see theorem\-~\ref{theo:tan} below.
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We have found that the simple equation reproduces this prediction, even when the potential is finite, when the density is asymptotically small (see Theorem\-~\ref{theo:tan}).
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To compute an approximation for $\mathfrak M(k)$, we follow the same procedure as above, which leads us to the following equation: for $k\neq0$,
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\begin{equation}
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(-\Delta+4e_\mu) u_\mu=(1-u_\mu)v+2\rho e_\mu u_\mu\ast u_\mu
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@ -288,7 +288,7 @@ and
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\label{gamma1}
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\end{equation}
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Differentiating \eqref{simpleq_momentum} leads once more to the operator $\mathfrak{K}_\mu$.
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Differentiating \eqref{simpleq_momentum} leads once more to the operator $\mathfrak{K}_{e_\mu}$.
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Therefore, a significant part of the analysis in this paper is aimed at understanding the operator $\mathfrak K_e$, as well as properties of solutions $u$ of the simple equation.
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Consider for example the problem of showing that $\rho'(e) > 0$ using the formula in \eqref{rpratio}. We will need to have $L^p$ to $L^q$ mapping properties of $\mathfrak{K}_e$, among other things, but all $L^p$ bounds on solutions $u$ of the simple equation system.
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@ -297,11 +297,11 @@ Integrating both sides of the simple equation, one sees that all solutions of th
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\int u(x)\ dx=\frac1\rho \ .
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\end{equation}
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Then since all physical solutions (those satisfying $u(x) \leqslant 1$) satisfy $0 \leqslant u(x) \leqslant 1$, it follows that $u\in L^p(\R^3)$ for all $1 \leqslant p \leqslant \infty$, and the obvious estimate that follows from this information is
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$\|u\|_p \leqslant \rho^{-1/p}$. However, one can do significantly better. We shall prove:
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$\|u\|_p \leqslant \rho^{-1/p}$. However, one can do significantly better. We shall prove the following lemma in section~\ref{UB}.
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\begin{lemma}\label{ul2b} For $1\leqslant p < 3$, solutions $u$ of \eqref{simp} satisfy
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\begin{equation}\label{sim6B}
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\|u\|_p \leqslant C_p e^{-(p-3/2p}\qquad{\rm where}\qquad C_p := 2(4\pi)^{1/p -1} \Gamma^{1/p}(3-p) (2p)^{(p-3)/p}\|v\|_1 \ .
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\|u\|_p \leqslant C_p e^{-(p-3)/2p}\qquad{\rm where}\qquad C_p := 2(4\pi)^{1/p -1} \Gamma^{1/p}(3-p) (2p)^{(p-3)/p}\|v\|_1 \ .
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\end{equation}
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In particular,
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\begin{equation}\label{sim6}
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@ -349,13 +349,13 @@ We shall also need various $L^p$ bounds on $u'$, and for these we need a detaile
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The next two theorems concern the monotonicity of $\rho\mapsto e(\rho)$ and convexity if $\rho\mapsto\rho e(\rho)$.
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These were conjectured in\-~\cite{CJL20}, and here, we proved them for small density $\rho$ (and, in the case of the monotonicity, also for large density).
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These were conjectured in\-~\cite{CJL20}, and here, we prove them for small density $\rho$ (and, in the case of the monotonicity, also for large density).
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\begin{theorem}[Monotonicity]\label{Mon} Assume that $(1+ |x|^4)v(x)\in L^1(\mathbb R^3)\cap L^2(\mathbb R^3)$.
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For
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$$
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e < e_\star:=\frac{\sqrt 2 \pi^{3}}{\|v\|_1^2}\qquad{\rm and\ for}\qquad e > \frac{2^3\|v\|_2^4}{\pi^4} $$
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$\rho(e)$ is strictly monotone increasing in $e$, and in these intervals $\rho(e)$ is continuously differentiable. If $u(e,\cdot)$ denotes the solution as a function of $e$, $u(e,\cdot)$ is continuously differentiable in $L^2(\R^3)$. Moreover,
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$\rho(e)$ is strictly monotone increasing in $e$, and in these intervals $\rho(e)$ is continuously differentiable. If $u(e,\cdot)$ denotes the solution of (\ref{simp}) as a function of $e$, $u(e,\cdot)$ is continuously differentiable in $L^2(\R^3)$. Moreover,
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\begin{equation}\label{rhopb2}
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\mathrm{for}\ e < e_\star\equiv\frac{\sqrt 2 \pi^{3}}{\|v\|_1^2}
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\quad\mathrm{we\ have}\quad
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@ -457,10 +457,10 @@ On account of \eqref{intu}, $\rho u$ is a probability density, and
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\begin{equation}
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0 \leqslant 4e(I - C_{\rho u}) \leqslant 4e
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\end{equation}
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so that $\mathfrak K_e$ is an unbounded operator. However, $e^{t4e(I - C_{\rho u})}$ is easily seen to be a positivity preserving contraction semigroup on $L^p$ for all $p$, as is $e^{t(\Delta + v)}$. Then by the Trotter Product formula, so is
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$e^{t(\Delta + v + 4e(1 - C_{\rho u})}$. Since
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so that $\mathfrak K_e$, as an operator from $L^1(\mathbb R^3)$ to $L^1(\mathbb R^3)$, is unbounded. However, $e^{t4e(I - C_{\rho u})}$ is easily seen to be a positivity preserving contraction semigroup on $L^p$ for all $p$, as is $e^{t(-\Delta + v)}$~\cite{Ne64,RS75b}. Then by the Trotter Product formula, so is
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$e^{t(-\Delta + v + 4e(1 - C_{\rho u})}$. Since
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\begin{equation}\label{intrep}
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\mathfrak{K}_e = \int_0^\infty dt e^{t(\Delta + v + 4e(1 - C_{\rho u})}\ ,
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\mathfrak{K}_e = \int_0^\infty dt e^{t(-\Delta + v + 4e(1 - C_{\rho u})}\ ,
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\end{equation}
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$\mathfrak{K}_e$ has a positive kernel denoted $\mathfrak{K}_e(x,y)$. We also define the convolution operator
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\begin{equation}\label{sim76}
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$$
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Fourier transforming the simple equation, one finds
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\begin{equation}\label{uhat}
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\rho\hat u(k)=\frac{k^2}{4e}+1-\sqrt{\left(\frac{k^2}{4e}+1\right)^2-\frac\rho{2e}\hat S(k)}
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\rho\widehat u(k)=\frac{k^2}{4e}+1-\sqrt{\left(\frac{k^2}{4e}+1\right)^2-\frac\rho{2e}\widehat S(k)}
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,\quad
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\hat S(k):=\int dx\ e^{ikx}(1-u(x))v(x)\ .
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\widehat S(k):=\int dx\ e^{ikx}(1-u(x))v(x)\ .
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\end{equation}
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By \eqref{intu}, $\rho \widehat{u}(0) =1$ and by the second equation in \eqref{simp}, $\frac{\rho}{2e}\widehat{S}(0) =1$,
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and from here one obtains
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\begin{equation}\label{facA}
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\left( k^2 + 4e(1 - \rho \widehat{u}(k)) \right)^{-1} \leqslant |k|^{-1}\left(k^2 + 2\sqrt{2e}\right)^{-1/2}
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\end{equation}
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The right side is square integrable, and in this way we obtain a bound on $\mathfrak{Y}_e$, and hence on $\mathfrak{K}_e$, from $L^1(\R^3)$ to $L^2(\R^3)$. The following lemma summarizes information that we obtain on $\mathfrak{K}_e$ that suffices to prove Theorem~\ref{Mon} on monotonicity.
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The right side is square integrable, and in this way we obtain a bound on $\mathfrak{Y}_e$ from $L^1(\R^3)$ to $L^2(\R^3)$. The following lemma (proved in section~\ref{UB}) summarizes information that we obtain on $\mathfrak{K}_e$ that suffices to prove Theorem~\ref{Mon} on monotonicity.
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@ -524,9 +524,9 @@ e\mapsto \int_{\R^3} \phi(x) ( \mathfrak{K}_e \psi)(x)\ dx
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\rho^{2}\int_{\R^3} (\mathfrak{K}_e v) u'*u'\ dx
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\end{equation}
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which shows that for small $\rho$, this is negligible compared to $\rho^{-2}$.
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By Young's inequality for convolutions, if we have bounds on $\|\mathfrak{K}_e v\|_p$ and $\|u'\|_q$ with $1/p + 2/q = 2$, we can bound the integral in \eqref{toughnut}. As we shall see below, since $v\geqslant 0$, $\mathfrak{K}_e v$ can decay at infinity no faster than $|x|^{-2}$, and hence cannot belong to $L^p$ for $p \leqslant 3/2$. Therefore, we will need to have a bound on $\|u'\|_q$ for fairly small $q$. We shall see that $\|u'\|_q < \infty$ for all $q>1$, and we shall obtain a bound on $\|u'\|_{4/3}$ that can be combined with our bound on $\|\mathfrak{K}_e v\|_2$ to obtain the necessary control on the integral in \eqref{toughnut}.
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By Young's inequality for convolutions, if we have bounds on $\|\mathfrak{K}_e v\|_p$ and $\|u'\|_q$ with $1/p + 2/q = 2$, we can bound the integral in \eqref{toughnut}. As we shall see below, since $v\geqslant 0$, $\mathfrak{K}_e v$ can decay at infinity no faster than $|x|^{-2}$, and hence cannot belong to $L^p$ for $p \leqslant 3/2$. Therefore, we will need to have a bound on $\|u'\|_q$ for fairly small $q$. We shall see that $\|u'\|_q < \infty$ for all $q>1$ (see Theorem~\ref{Lpu'}), and we shall obtain a bound on $\|u'\|_{4/3}$ that can be combined with our bound on $\|\mathfrak{K}_e v\|_2$ to obtain the necessary control on the integral in \eqref{toughnut}.
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To do this, we need something more incisive than the bound \eqref{facA}. We shall show that
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To do this, we need something more incisive than the bound \eqref{facA}. We shall show (see section\-~\ref{sec:uprime}) that
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$\mathfrak{Y}_e$, factors as the product of three commuting operators operators
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\begin{equation}\label{3fac}
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\mathfrak{Y}_e = (-\Delta)^{-1/2} (-\Delta+ 8e)^{-1/2} [I + \mathfrak{H}_e]
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@ -614,53 +614,53 @@ This provides the control on $\|u'\|_q$ that we need to prove the theorem on co
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\end{equation}
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in terms of which (\ref{uhat}) becomes
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\begin{equation}
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\rho\hat u=(\kappa^2+1)\left(1-\sqrt{1-\frac{\frac\rho{2e}\hat S}{(\kappa^2+1)^2}}\right)
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\rho\widehat u=(\kappa^2+1)\left(1-\sqrt{1-\frac{\frac\rho{2e}\widehat S}{(\kappa^2+1)^2}}\right)
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.
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\label{uhat_factorized}
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\end{equation}
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For small $\kappa$, since $x^4v$ is integrable, $\hat S$ is $\mathcal C^4$
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For small $\kappa$, since $x^4v$ is integrable, $\widehat S$ is $\mathcal C^4$
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\begin{equation}
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\frac\rho{2e}\hat S=1-\beta \kappa^2+O(e^2\kappa^4)
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\frac\rho{2e}\widehat S=1-\beta \kappa^2+O(e^2\kappa^4)
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\label{expS}
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\end{equation}
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and $\beta$ is defined in\-~(\ref{betadef}):
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\begin{equation}\label{betabound}
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\beta = -\frac\rho{4e}\partial_\kappa^2 \hat S \leqslant \rho\|x^2v\|_1
|
||||
\beta = -\frac\rho{4e}\partial_\kappa^2 \widehat S \leqslant \rho\|x^2v\|_1
|
||||
.
|
||||
\end{equation}
|
||||
Therefore, defining
|
||||
\begin{equation}
|
||||
\hat U_1:=(\kappa^2+1)^{-2}\left(1-\sqrt{1-\frac{(1-\beta \kappa^2)}{(\kappa^2+1)^2}}\right)
|
||||
\widehat U_1:=(\kappa^2+1)^{-2}\left(1-\sqrt{1-\frac{(1-\beta \kappa^2)}{(\kappa^2+1)^2}}\right)
|
||||
\end{equation}
|
||||
$\hat U_1$ coincides with $\hat u$ asymptotically as $\kappa\to0$
|
||||
and we chose the prefactor $(\kappa^2+1)^{-2}$ in such a way that $\hat U_1$ is integrable.
|
||||
$\widehat U_1$ coincides with $\widehat u$ asymptotically as $\kappa\to0$
|
||||
and we chose the prefactor $(\kappa^2+1)^{-2}$ in such a way that $\widehat U_1$ is integrable.
|
||||
Define the remainder term
|
||||
\begin{equation}
|
||||
\hat U_2:=\rho\hat u-\hat U_1
|
||||
\widehat U_2:=\rho\widehat u-\widehat U_1
|
||||
=(\kappa^2+1)\left(1-\sqrt{1-2\zeta_1}\right)-(\kappa^2+1)^{-2}\left(1-\sqrt{1-2\zeta_2}\right)
|
||||
\end{equation}
|
||||
with
|
||||
\begin{equation}
|
||||
\zeta_1:=\frac{\frac\rho{4e}\hat S}{(\kappa^2+1)^2}
|
||||
\zeta_1:=\frac{\frac\rho{4e}\widehat S}{(\kappa^2+1)^2}
|
||||
,\quad
|
||||
\zeta_2:=\frac{1-\beta \kappa^2}{2(\kappa^2+1)^2}
|
||||
.
|
||||
\label{zetas}
|
||||
\end{equation}
|
||||
The rest of the proof proceeds as follows: we show that the Fourier transform of $\hat U_1$ decays like $|x|^{-4}$ by direct analysis, then we show that $\Delta^2\hat U_2$ is integrable and square integrable, which implies that it is subdominant as $|x|\to\infty$.
|
||||
The rest of the proof proceeds as follows: we show that the Fourier transform of $\widehat U_1$ decays like $|x|^{-4}$ by direct analysis, then we show that $\Delta^2\widehat U_2$ is integrable and square integrable, which implies that it is subdominant as $|x|\to\infty$.
|
||||
|
||||
|
||||
\point We compute $U_1(x):=\int\frac{dk}{(2\pi)^3}e^{-ikx}\hat U_1(k)$.
|
||||
\point We compute $U_1(x):=\int\frac{dk}{(2\pi)^3}e^{-ikx}\widehat U_1(k)$.
|
||||
We write
|
||||
\begin{equation}
|
||||
\sqrt{1 - \frac{1-\beta \kappa^2}{(1+\kappa^2)^2}} = \frac{\kappa}{1+\kappa^2} \sqrt{2 + \beta + \kappa^2} = \frac{1}{\pi} \frac{|\kappa|(2 + \beta + \kappa^2)}{1+\kappa^2} \int_0^\infty \frac{1}{2+\beta + t + \kappa^2} t^{-1/2} dt \ \ .
|
||||
\end{equation}
|
||||
Therefore,
|
||||
\begin{equation}
|
||||
\hat{U}_1 := (\kappa^2+1)^{-2} - \frac{\kappa}{\pi} (\kappa^2+1)^{-2} \left(1 + (\beta+1)\frac{1}{1+\kappa^2}\right) \int_0^\infty \frac{1}{2+\beta + t + \kappa^2} t^{-1/2} dt
|
||||
\widehat{U}_1 := (\kappa^2+1)^{-2} - \frac{\kappa}{\pi} (\kappa^2+1)^{-2} \left(1 + (\beta+1)\frac{1}{1+\kappa^2}\right) \int_0^\infty \frac{1}{2+\beta + t + \kappa^2} t^{-1/2} dt
|
||||
.
|
||||
\end{equation}
|
||||
We take the inverse Fourier transform of $\hat U_1$, recalling the definition of $\kappa$\-~(\ref{kappa})
|
||||
We take the inverse Fourier transform of $\widehat U_1$, recalling the definition of $\kappa$\-~(\ref{kappa})
|
||||
\begin{equation}
|
||||
U_1(x)=\frac{e^{\frac32}}\pi e^{-2\sqrt e|x|}
|
||||
-\frac1\pi \left(\delta(x)+\frac{(\beta+1)e}\pi\frac{e^{-2\sqrt e|x|}}{|x|}\right)
|
||||
|
|
@ -711,7 +711,7 @@ This provides the control on $\|u'\|_q$ that we need to prove the theorem on co
|
|||
\end{equation}
|
||||
|
||||
\point
|
||||
We now show that $\Delta^2\hat U_2$ is integrable and square-integrable.
|
||||
We now show that $\Delta^2\widehat U_2$ is integrable and square-integrable.
|
||||
We use the fact that
|
||||
\begin{equation}
|
||||
16e^2\Delta^2\equiv\partial_\kappa^4+\frac4\kappa\partial_\kappa^3
|
||||
|
|
@ -719,7 +719,7 @@ This provides the control on $\|u'\|_q$ that we need to prove the theorem on co
|
|||
\end{equation}
|
||||
We have, by the Leibniz rule,
|
||||
\begin{equation}
|
||||
\partial_\kappa^n\hat U_2=
|
||||
\partial_\kappa^n\widehat U_2=
|
||||
\sum_{i=0}^n{n\choose i}\left(
|
||||
\partial_\kappa^{n-i}(\kappa^2+1)\partial_\kappa^i(1-\sqrt{1-2\zeta_1})
|
||||
-
|
||||
|
|
@ -737,13 +737,13 @@ This provides the control on $\|u'\|_q$ that we need to prove the theorem on co
|
|||
\label{dsqrt}
|
||||
\end{equation}
|
||||
for some family of constants $c_{l_1,\cdots,l_p}^{(p,n)}$ which can easily be computed explicitly, but this is not needed.
|
||||
Now, since $S\geqslant 0$, $\frac\rho{1e}|\hat S|\leqslant 1$, so $|\zeta_1|\leqslant\frac12$ and $\zeta_1=\frac12$ if and only if $\kappa=0$.
|
||||
Therefore, $\hat U_2$ is bounded when $\kappa$ is away from 0, so it suffices to show that $\Delta^2\hat U_2$ is integrable and square integrable at infinity and at 0.
|
||||
Now, since $S\geqslant 0$, $\frac\rho{1e}|\widehat S|\leqslant 1$, so $|\zeta_1|\leqslant\frac12$ and $\zeta_1=\frac12$ if and only if $\kappa=0$.
|
||||
Therefore, $\widehat U_2$ is bounded when $\kappa$ is away from 0, so it suffices to show that $\Delta^2\widehat U_2$ is integrable and square integrable at infinity and at 0.
|
||||
|
||||
|
||||
\subpoint We first consider the behavior at infinity, and assume that $\kappa$ is sufficiently large.
|
||||
The fact that $\partial_\kappa^{n-i}(\kappa^2+1)^{-2}\partial_\kappa^i(1-\sqrt{1-2\zeta_2})$ is integrable and square integrable at infinity follows immediately from\-~(\ref{zetas}).
|
||||
To prove the corresponding claim for $\zeta_1$, we use the fact that $|x|^4 v$ square integrable, which implies that $\hat S$ is as well.
|
||||
To prove the corresponding claim for $\zeta_1$, we use the fact that $|x|^4 v$ square integrable, which implies that $\widehat S$ is as well.
|
||||
Therefore, by\-~(\ref{zetas}) for $0\leqslant n\leqslant 4$, $\kappa^2\partial_\kappa^n\zeta_1$ is integrable at infinity, and, therefore, square-integrable at infinity.
|
||||
Furthermore, by\-~(\ref{zetas}), $\zeta_1<\frac12-\epsilon$ for large $\kappa$, and $\partial^n\zeta_1$ is bounded, so $\partial_\kappa^{n-i}(\kappa^2+1)\partial_\kappa^i(1-\sqrt{1-2\zeta_1})$ is integrable and square integrable.
|
||||
|
||||
|
|
@ -789,12 +789,12 @@ This provides the control on $\|u'\|_q$ that we need to prove the theorem on co
|
|||
\end{equation}
|
||||
Thus, by\-~(\ref{leibnitz}),
|
||||
\begin{equation}
|
||||
|\partial_\kappa^4\hat U_2|=O(\kappa^{-1})
|
||||
|\partial_\kappa^4\widehat U_2|=O(\kappa^{-1})
|
||||
,\quad
|
||||
\frac4\kappa|\partial_\kappa^3\hat U_2|=O(\kappa^{-1})
|
||||
\frac4\kappa|\partial_\kappa^3\widehat U_2|=O(\kappa^{-1})
|
||||
.
|
||||
\end{equation}
|
||||
Thus, $\Delta^2\hat U_2$ is integrable and square integrable.
|
||||
Thus, $\Delta^2\widehat U_2$ is integrable and square integrable.
|
||||
And since the $O(\cdot)$ hold uniformly in $e$ on all compact sets, $U_2|x|^4$ is bounded and square integrable uniformly in $e$ on all compact sets.
|
||||
\qed
|
||||
|
||||
|
|
@ -1127,21 +1127,21 @@ in which $\mathfrak Y_e$ is defined in \eqref{sim76}.
|
|||
uniformly in $x$.
|
||||
We work in Fourier space: by \eqref{uhat},
|
||||
\begin{equation}
|
||||
\hat\xi(2\sqrt{e}k)=
|
||||
\widehat\xi(2\sqrt{e}k)=
|
||||
\frac1{4e}
|
||||
\left(\frac{k^2+1}{\sqrt{(k^2+1)^2-\frac\rho{2e}\hat S(2\sqrt{e}k)}}-1\right)
|
||||
\left(\frac{k^2+1}{\sqrt{(k^2+1)^2-\frac\rho{2e}\widehat S(2\sqrt{e}k)}}-1\right)
|
||||
.
|
||||
\label{x1rescale}
|
||||
\end{equation}
|
||||
Since $S(x)\geqslant 0$, $|\hat S(k)|\leqslant|\hat S(0)|=\frac{2e}\rho$, and since $S$ is symmetric, $\hat S$ is real, so
|
||||
Since $S(x)\geqslant 0$, $|\widehat S(k)|\leqslant|\widehat S(0)|=\frac{2e}\rho$, and since $S$ is symmetric, $\widehat S$ is real, so
|
||||
\begin{equation}
|
||||
|\hat\xi(2\sqrt{e}k)|\leqslant
|
||||
|\widehat\xi(2\sqrt{e}k)|\leqslant
|
||||
\frac1{4e}
|
||||
\left(\frac{k^2+1}{\sqrt{(k^2+1)^2-1}}-\frac{k^2+1}{\sqrt{(k^2+1)^2+1}}\right)
|
||||
\label{dominatedxi1}
|
||||
\end{equation}
|
||||
which is integrable.
|
||||
Next, note that $\frac\rho{2e}\hat S(2\sqrt e k)\to1$ and
|
||||
Next, note that $\frac\rho{2e}\widehat S(2\sqrt e k)\to1$ and
|
||||
\begin{equation}\label{idint}
|
||||
\int\left(\frac{k^2+1}{\sqrt{(k^2+1)^2-1}}-1\right)\ \frac{dk}{8\pi^3}=\frac1{3\pi^2\sqrt2}
|
||||
\end{equation}
|
||||
|
|
@ -1153,13 +1153,13 @@ Next, by\-~(\ref{x1rescale}) and\-~(\ref{idint}),
|
|||
=&
|
||||
\frac{\sqrt{e}}{4\pi^3} \int (e^{-i2\sqrt{e}kx}-1)
|
||||
\left(
|
||||
\frac{k^2+1}{\sqrt{(k^2+1)^2-\frac \rho{2e}\hat S(2\sqrt{e}k)}}-1
|
||||
\frac{k^2+1}{\sqrt{(k^2+1)^2-\frac \rho{2e}\widehat S(2\sqrt{e}k)}}-1
|
||||
\right)\ dk
|
||||
\\[0.5cm]
|
||||
&+
|
||||
\frac{\sqrt{e}}{4\pi^3}\int
|
||||
\left(
|
||||
\frac{k^2+1}{\sqrt{(k^2+1)^2-\frac \rho{2e}\hat S(2\sqrt{e}k)}}
|
||||
\frac{k^2+1}{\sqrt{(k^2+1)^2-\frac \rho{2e}\widehat S(2\sqrt{e}k)}}
|
||||
-\frac{k^2+1}{\sqrt{(k^2+1)^2-1}}
|
||||
\right)\ dk
|
||||
.
|
||||
|
|
@ -1169,25 +1169,25 @@ By\-~(\ref{dominatedxi1}), the first integrand is absolutely integrable, so
|
|||
\begin{equation}
|
||||
\frac{\sqrt{e}}{4\pi^3} \int (e^{-i2\sqrt{e}kx}-1)
|
||||
\left(
|
||||
\frac{k^2+1}{\sqrt{(k^2+1)^2-\frac \rho{2e}\hat S(2\sqrt{e}k)}}-1
|
||||
\frac{k^2+1}{\sqrt{(k^2+1)^2-\frac \rho{2e}\widehat S(2\sqrt{e}k)}}-1
|
||||
\right)\ dk
|
||||
=o(\sqrt e)
|
||||
\end{equation}
|
||||
uniformly in $x$.
|
||||
Furthermore, since $\frac\rho{2e}\hat S\leqslant 1$ and $1-(1+\epsilon)^{-\frac12}\leqslant\frac\epsilon 2$ for all $\epsilon\geqslant 0$,
|
||||
Furthermore, since $\frac\rho{2e}\widehat S\leqslant 1$ and $1-(1+\epsilon)^{-\frac12}\leqslant\frac\epsilon 2$ for all $\epsilon\geqslant 0$,
|
||||
\begin{equation}
|
||||
\left|
|
||||
\frac{k^2+1}{\sqrt{(k^2+1)^2-\frac \rho{2e}\hat S(2\sqrt{e}k)}}
|
||||
\frac{k^2+1}{\sqrt{(k^2+1)^2-\frac \rho{2e}\widehat S(2\sqrt{e}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}\hat S(2\sqrt{e}k)}2
|
||||
\frac{k^2+1}{((k^2+1)^2-1)^{\frac32}}\frac{1-\frac\rho{2e}\widehat S(2\sqrt{e}k)}2
|
||||
\end{equation}
|
||||
and since $\hat S$ is the Fourier transform of $(1-u)v$ which is absolutely integrable, $\hat S$ is uniformly continuous, so
|
||||
and since $\widehat S$ is the Fourier transform of $(1-u)v$ which is absolutely integrable, $\widehat S$ is uniformly continuous, so
|
||||
\begin{equation}
|
||||
\frac{\sqrt{e}}{4\pi^3}\int dk\
|
||||
\left|
|
||||
\frac{k^2+1}{\sqrt{(k^2+1)^2-\frac \rho{2e}\hat S(2\sqrt{e}k)}}
|
||||
\frac{k^2+1}{\sqrt{(k^2+1)^2-\frac \rho{2e}\widehat S(2\sqrt{e}k)}}
|
||||
-\frac{k^2+1}{\sqrt{(k^2+1)^2-1}}
|
||||
\right|
|
||||
=o(\sqrt{e})
|
||||
|
|
@ -1261,7 +1261,7 @@ We now turn to the proof of\-~(\ref{M_asym}).
|
|||
First of all, by\-~(\ref{neglect_denom}),
|
||||
\begin{equation}
|
||||
\mathfrak M(k)=
|
||||
\rho\hat u_0(k)(1+O(\rho))\int v(x)\mathfrak K_e\cos(k\cdot x)\ dx
|
||||
\rho\widehat u_0(k)(1+O(\rho))\int v(x)\mathfrak K_e\cos(k\cdot x)\ dx
|
||||
\end{equation}
|
||||
Proceeding as in section\-~\ref{sec:condensate_fraction}, we use the resolvent identity to rewrite
|
||||
\begin{equation}
|
||||
|
|
@ -1275,7 +1275,7 @@ in which $\mathfrak Y_e$ is defined in \eqref{sim76}, so
|
|||
\begin{equation}
|
||||
\int v(x)\mathfrak K_e\cos(kx)\ dx
|
||||
=
|
||||
\frac{\hat v(k)-\int e^{ikx}v\mathfrak K_e v\ dx}{k^2+4e(1-\rho \hat u(k))}
|
||||
\frac{\widehat v(k)-\int e^{ikx}v\mathfrak K_e v\ dx}{k^2+4e(1-\rho \widehat u(k))}
|
||||
.
|
||||
\end{equation}
|
||||
Since $v\mathfrak K_e \leqslant v(-\Delta+v)^{-1}v$ which is integrable: by\-~(\ref{intscat})
|
||||
|
|
@ -1288,7 +1288,7 @@ we have, by dominated convergence, in the limit $e\to0$ and $|k|\to0$,
|
|||
\begin{equation}
|
||||
\int v(x)\mathfrak K_e\cos(kx)\ dx
|
||||
\sim
|
||||
\frac{4\pi a_0}{k^2+4e(1-\rho\hat u(k))}
|
||||
\frac{4\pi a_0}{k^2+4e(1-\rho\widehat u(k))}
|
||||
\end{equation}
|
||||
so, as $\kappa\to\infty$,
|
||||
\begin{equation}
|
||||
|
|
@ -1378,21 +1378,21 @@ Let $\mathfrak{Y}_e$ be defined as in \eqref{sim76}. Then the positive kernels o
|
|||
$$ 0 \leqslant \mathfrak{K}_e \psi \leqslant \mathfrak{Y}_e\psi \qquad{\rm and\ hence}\qquad \|\mathfrak{K}_e \psi\|_2 \leqslant \|\mathfrak{Y}_e \psi\|_2 \ .$$
|
||||
Then by \eqref{sim76}, for non-negative $\psi\in L^1$, $\mathfrak{Y}_e\psi\in L^2(\R^3)$ with
|
||||
\begin{equation}\label{sim30}\
|
||||
\|\mathfrak{Y}_e\psi \|_2^2 \leqslant \frac{\|\psi\|_1^2}{(2\pi)^3} \int dk [k^2 +4e (1- \rho\hat u(k))]^{-2} \ .
|
||||
\|\mathfrak{Y}_e\psi \|_2^2 \leqslant \frac{\|\psi\|_1^2}{(2\pi)^3} \int dk [k^2 +4e (1- \rho\widehat u(k))]^{-2} \ .
|
||||
\end{equation}
|
||||
|
||||
|
||||
|
||||
Recall from \eqref{uhat} that
|
||||
$$\rho \hat{u}(k) = 1 + \frac{k^2}{4e} - \sqrt{ \left(1 + \frac{k^2}{4e}\right)^2 -\frac{\rho}{2e}\hat{S}(k)}\quad{\rm where}\quad \hat{S}(k) = \int v(1-u)e^{-ikx} dx\ ,$$
|
||||
$$\rho \widehat{u}(k) = 1 + \frac{k^2}{4e} - \sqrt{ \left(1 + \frac{k^2}{4e}\right)^2 -\frac{\rho}{2e}\widehat{S}(k)}\quad{\rm where}\quad \widehat{S}(k) = \int v(1-u)e^{-ikx} dx\ ,$$
|
||||
and hence
|
||||
$$(1- \rho \hat{u}(k)) \geqslant \sqrt{ \left(1 + \frac{k^2}{4e}\right)^2 -\frac{\rho}{2e}\hat{S}(k)}\ .$$
|
||||
$$(1- \rho \widehat{u}(k)) \geqslant \sqrt{ \left(1 + \frac{k^2}{4e}\right)^2 -\frac{\rho}{2e}\widehat{S}(k)}\ .$$
|
||||
By \eqref{simp} $\frac{\rho}{2e}S(k) \leqslant 1$ and hence
|
||||
${\displaystyle
|
||||
4e(1- \rho \hat{u}(k)) \geqslant \sqrt{k^4 + 8e k^2} \geqslant \sqrt{8e}|k|}$.
|
||||
4e(1- \rho \widehat{u}(k)) \geqslant \sqrt{k^4 + 8e k^2} \geqslant \sqrt{8e}|k|}$.
|
||||
Therefore
|
||||
$$
|
||||
\int [k^2 +4e (1- \rho\hat u(k))]^{-2}dk \leqslant \int dk [k^2 + \sqrt{8e}|k|]^{-2} = 4\pi \int_0^\infty dr \frac{r^2}{[r^2 + \sqrt{8e} r]^2} =\frac{2\pi}{\sqrt{2e}}\ .
|
||||
\int [k^2 +4e (1- \rho\widehat u(k))]^{-2}dk \leqslant \int dk [k^2 + \sqrt{8e}|k|]^{-2} = 4\pi \int_0^\infty dr \frac{r^2}{[r^2 + \sqrt{8e} r]^2} =\frac{2\pi}{\sqrt{2e}}\ .
|
||||
$$
|
||||
For the final part, we apply the bound just proved to the positive and negative parts of $\psi$ separately.
|
||||
\end{proof}
|
||||
|
|
@ -1496,7 +1496,7 @@ Then $L_{p,q}$ consists of the measurable functions $f$ such that $\|f\|_{p,q} <
|
|||
\begin{equation}\label{isias72}
|
||||
|\{x\ :\ |f(x)| > \lambda\}| \leqslant C\lambda^{-p}\ .
|
||||
\end{equation}
|
||||
That is $L_{p,\infty}$ is weak $L^p$ and hence $L_{p.\infty} \subset L^p$.
|
||||
That is $L_{p,\infty}$ is weak $L^p$ and hence $L_{p,\infty} \subset L^p$.
|
||||
and in this case,
|
||||
$$
|
||||
\|f\|_{p,\infty} \leqslant C^{1/p}
|
||||
|
|
@ -1700,7 +1700,7 @@ $$
|
|||
\end{proof}
|
||||
|
||||
|
||||
\section{Bounds on $u'$ -- Proof of Theorem~\ref{Lpu'} } Recall from \eqref{upform} that
|
||||
\section{Bounds on $u'$ -- Proof of Theorem~\ref{Lpu'} }\label{sec:uprime} Recall from \eqref{upform} that
|
||||
\begin{equation}\label{bamb34}
|
||||
u' = \mathfrak{K}_e \varphi \qquad{\rm where }\quad \varphi = 2\left( 1 + \rho'\frac{e}{\rho}\right)\rho u*u -4u\ ,
|
||||
\end{equation}
|
||||
|
|
@ -1753,23 +1753,23 @@ It then follows that for $3/2 \leqslant q \leqslant \infty$,
|
|||
|
||||
The operator $\mathfrak{Y}_e$ is also a convolution operator, but somewhat more complicated. It has a useful factorization that we now describe.
|
||||
|
||||
Note that the Fourier transform renders $\mathfrak{Y}_e$ as the operation of multiplication by $[k^2 + 4e(1- \rho \hat{u}(k))]^{-1}$.
|
||||
Note that the Fourier transform renders $\mathfrak{Y}_e$ as the operation of multiplication by $[k^2 + 4e(1- \rho \widehat{u}(k))]^{-1}$.
|
||||
Recall that
|
||||
$$\rho \hat{u}(k) = 1 + \frac{k^2}{4e} - \sqrt{ \left(1 + \frac{k^2}{4e}\right)^2 -\frac{\rho}{2e}\hat{S}(k)}\ ,$$
|
||||
$$\rho \widehat{u}(k) = 1 + \frac{k^2}{4e} - \sqrt{ \left(1 + \frac{k^2}{4e}\right)^2 -\frac{\rho}{2e}\widehat{S}(k)}\ ,$$
|
||||
where
|
||||
$\hat{S}(k) = \int v(1-u)e^{-ikx} dx$, $\frac{\rho}{2e}S(k) \leqslant 1$ and hence
|
||||
$\widehat{S}(k) = \int v(1-u)e^{-ikx} dx$, $\frac{\rho}{2e}S(k) \leqslant 1$ and hence
|
||||
\begin{equation}\label{sim40}
|
||||
[k^2 + 4e(1- \rho \hat{u}(k))]^{-1}
|
||||
[k^2 + 4e(1- \rho \widehat{u}(k))]^{-1}
|
||||
= \frac{1+\widehat H(k)}{k(k^2 + 8e)^{1/2}}
|
||||
\end{equation}
|
||||
with
|
||||
$$\widehat{H}(k) := \left(1 + \frac{16e^2(1-\frac{\rho}{2e}\hat{S}(k))}{ k^4 + 8ek^2 } \right)^{-1/2} -1.$$
|
||||
Since $\lim_{k\to\infty}\hat{S}(k) = 0$, $\widehat{H}$ is integrable.
|
||||
$$\widehat{H}(k) := \left(1 + \frac{16e^2(1-\frac{\rho}{2e}\widehat{S}(k))}{ k^4 + 8ek^2 } \right)^{-1/2} -1.$$
|
||||
Since $\lim_{k\to\infty}\widehat{S}(k) = 0$, $\widehat{H}$ is integrable.
|
||||
|
||||
It is more work to see that its inverse Fourier transform, $H(x)$ is also integrable, but we show this below. It turns out that $H(x)$ is not non-negative. Had this been the case we would have that
|
||||
$\|H\|_1 =\widehat{H}(0)$. To compute this, we expand
|
||||
\begin{equation}
|
||||
\frac{\rho}{2e}\hat{S}(k) = 1 + \beta k^2 + O(k^4) .
|
||||
\frac{\rho}{2e}\widehat{S}(k) = 1 + \beta k^2 + O(k^4) .
|
||||
\end{equation}
|
||||
Therefore,
|
||||
$$
|
||||
|
|
@ -1783,7 +1783,7 @@ and in particular, for a different $C$ still independent of $e$, $\| H\|_1 \leq
|
|||
\end{lemma}
|
||||
\begin{proof}[Proof of Lemma~\ref{Hlem}]
|
||||
Recall that
|
||||
$$\widehat{H}(k) := \left(1 + G(k) \right)^{-1/2} -1 \qquad{\rm where}\qquad G(k) := \frac{16e^2(1-\frac{\rho}{2e}\hat{S}(k))}{ |k|^4 + 8e|k|^2}\ .$$
|
||||
$$\widehat{H}(k) := \left(1 + G(k) \right)^{-1/2} -1 \qquad{\rm where}\qquad G(k) := \frac{16e^2(1-\frac{\rho}{2e}\widehat{S}(k))}{ |k|^4 + 8e|k|^2}\ .$$
|
||||
The proof is very similar to that of Theorem~\ref{theo:pointwise}, except simpler: One shows that $\widehat{H}$ and $\Delta^2 \widehat{H}$ are integrable, with $\|\widehat{H}\|_1 + \|\Delta^2 \widehat{H}\|_1 \leqslant Ce^{1/2}$. The claim now follows from the Riemann-Lebesgue lemma.
|
||||
\end{proof}
|
||||
|
||||
|
|
@ -1961,7 +1961,7 @@ By direct computations, we find that
|
|||
\rho=\frac1{\int dx\ u(x)}=\frac{b^3}{c\pi^2}
|
||||
\label{rho}
|
||||
,\quad
|
||||
\hat u(k)=\frac{\pi^2 c}{b^3}e^{-\frac{|k|}b}
|
||||
\widehat u(k)=\frac{\pi^2 c}{b^3}e^{-\frac{|k|}b}
|
||||
,\quad
|
||||
u\ast u=\frac{2\pi^2c^2}{b^3(4+b^2 x^2)^2}
|
||||
\end{equation}
|
||||
|
|
|
|||
|
|
@ -0,0 +1,9 @@
|
|||
v0.1:
|
||||
|
||||
* Added: references to [Nelson, 1964] and [Reed, Simon, 1975].
|
||||
|
||||
* Fixed: sign of Laplacian in definition of \mathfrak K_e.
|
||||
|
||||
* Added: references to proofs in introduction.
|
||||
|
||||
* Fixed: miscellaneous typos.
|
||||
|
|
@ -1,15 +1,16 @@
|
|||
@article{BBe18, doi = {10.1007/s00220-017-3016-5}, url = {https://doi.org/10.1007
|
||||
2Fs00220-017-3016-5}, year = 2017, month = {nov}, publisher = {Springer Nature}, volume = {359}, number = {3}, pages = {975--1026}, author = {Chiara Boccato and Christian Brennecke and Serena Cenatiempo and Benjamin Schlein}, title = {Complete Bose{\textendash}Einstein Condensation in the Gross{\textendash}Pitaevskii Regime}, journal = {Communications in Mathematical Physics} }
|
||||
@article{Bo47, title={On the theory of superfluidity}, author={Bogolyubov, Nikolay Nikolaevich}, journal={Izv. Akad. Nauk Ser. Fiz.}, volume={11}, pages={23--32}, year={1947} }
|
||||
@article{CHe, author = {Carlen, Eric A. and Holzmann, Markus and Jauslin, Ian and Lieb, Elliott H.},journal = {in preparation} }
|
||||
@article{CJL20, title={Analysis of a simple equation for the ground state energy of the Bose gas}, author = {Carlen, Eric A. and Jauslin, Ian and Lieb, Elliott H.},journal = {Pure and Applied Analysis, in press, arXiv preprint: 1912.04987}, year={2020} }
|
||||
@article{CHe, author = {Carlen, Eric A. and Holzmann, Markus and Jauslin, Ian and Lieb, Elliott H.},journal = {Physical Review A, in press}, eprint = {arXiv:2011.10869} }
|
||||
@article{CJL20, doi = {10.2140/paa.2020.2.659}, url = {https://doi.org/10.2140
|
||||
2Fpaa.2020.2.659}, year = 2020, month = {nov}, publisher = {Mathematical Sciences Publishers}, volume = {2}, number = {3}, pages = {659--684}, author = {Eric A. Carlen and Ian Jauslin and Elliott H. Lieb}, title = {Analysis of a simple equation for the ground state energy of the Bose gas}, journal = {Pure and Applied Analysis} }
|
||||
@article{CAL09, doi = {10.1103/physreva.79.053640}, url = {https://doi.org/10.1103
|
||||
2Fphysreva.79.053640}, year = 2009, month = {may}, publisher = {American Physical Society ({APS})}, volume = {79}, number = {5}, author = {R. Combescot and F. Alzetto and X. Leyronas}, title = {Particle distribution tail and related energy formula}, journal = {Physical Review A} }
|
||||
@article{Dy57, doi = {10.1103/physrev.106.20}, url = {https://doi.org/10.1103
|
||||
2Fphysrev.106.20}, year = 1957, month = {apr}, publisher = {American Physical Society ({APS})}, volume = {106}, number = {1}, pages = {20--26}, author = {F. J. Dyson}, title = {Ground-State Energy of a Hard-Sphere Gas}, journal = {Physical Review} }
|
||||
@article{ESY08, doi = {10.1103/physreva.78.053627}, url = {https://doi.org/10.1103
|
||||
2Fphysreva.78.053627}, year = 2008, month = {nov}, publisher = {American Physical Society ({APS})}, volume = {78}, number = {5}, author = {L{\'{a}}szl{\'{o}} Erd{\H{o}}s and Benjamin Schlein and Horng-Tzer Yau}, title = {Ground-state energy of a low-density Bose gas: A second-order upper bound}, journal = {Physical Review A} , pages={053627}}
|
||||
@article{FS20, title={The energy of dilute Bose gases}, author={Fournais, S\o{}ren and Solovej, Jan Philip}, journal={The Annals of Mathematics, in press, arXiv preprint:1904.06164}, year={2020} }
|
||||
@article{FS20, ISSN = {0003486X, 19398980}, URL = {https://www.jstor.org/stable/10.4007/annals.2020.192.3.5}, author = {S{\o}ren Fournais and Jan Philip Solovej}, journal = {Annals of Mathematics}, number = {3}, pages = {893--976}, publisher = {[Annals of Mathematics, Trustees of Princeton University on Behalf of the Annals of Mathematics, Mathematics Department, Princeton University]}, title = {The energy of dilute Bose gases}, volume = {192}, year = {2020}, doi = {10.4007/annals.2020.192.3.5} }
|
||||
@article{GS09, doi = {10.1007/s10955-009-9718-0}, url = {https://doi.org/10.1007
|
||||
2Fs10955-009-9718-0}, year = 2009, month = {apr}, publisher = {Springer Science and Business Media {LLC}}, volume = {135}, number = {5-6}, pages = {915--934}, author = {Alessandro Giuliani and Robert Seiringer}, title = {The Ground State Energy of the Weakly Interacting Bose Gas at High Density}, journal = {Journal of Statistical Physics} }
|
||||
@article{KLS88, doi = {10.1007/bf01023854}, url = {https://doi.org/10.1007
|
||||
|
|
@ -30,6 +31,8 @@
|
|||
@article{NE17, doi = {10.1088/1361-6633/aa50e8}, url = {https://doi.org/10.1088
|
||||
2F1361-6633
|
||||
2Faa50e8}, year = 2017, month = {mar}, publisher = {{IOP} Publishing}, volume = {80}, number = {5}, pages = {056001}, author = {Pascal Naidon and Shimpei Endo}, title = {Efimov physics: a review}, journal = {Reports on Progress in Physics}}
|
||||
@article{Ne64, doi = {10.1063/1.1704124}, url = {https://doi.org/10.1063
|
||||
2F1.1704124}, year = 1964, month = {mar}, publisher = {{AIP} Publishing}, volume = {5}, number = {3}, pages = {332--343}, author = {Edward Nelson}, title = {Feynman Integrals and the Schrödinger Equation}, journal = {Journal of Mathematical Physics}}
|
||||
@article{On63, doi = {10.1215/s0012-7094-63-03015-1}, url = {https://doi.org/10.1215
|
||||
2Fs0012-7094-63-03015-1}, year = 1963, month = {mar}, publisher = {Duke University Press}, volume = {30}, number = {1}, pages = {129--142}, author = {Richard O'Neil}, title = {Convolution operators and $L(p,q)$ spaces}, journal = {Duke Mathematical Journal} }
|
||||
@book{RS75b, title={Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness}, author={Reed, Michael and Simon, Barry}, edition={2}, year={1975}, publisher={Academic Press, New York}}
|
||||
|
|
|
|||
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Reference in New Issue