Update to v0.2

Added: Discussion of uniqueness of the solution of the Simple Equation

  Added: Exercise: prove \psi_0>=0 using |\psi_0|

  Added: Links in bibliography

  Removed: Commented blocks

  Fixed: Formatting

Changelog

@@ -0,0 +1,20 @@
+v0.2:
+  * Added: Discussion of uniqueness of the solution of the Simple Equation
+
+  * Added: Exercise: prove \psi_0>=0 using |\psi_0|
+
+  * Added: Links in bibliography
+
+  * Removed: Commented blocks
+
+  * Fixed: Formatting
+
+
+v0.1:
+
+  * Fixed: H_N is not compact, e^{-H_N} is.
+
+  * Fixed: Removed confusing discussion of Anyons
+
+  * Added: Discussion of compactness.
+

Jauslin_2023c.tex

@@ -269,6 +269,7 @@ The microscopic state of a system of quantum particles is given by a wavefunctio
   \label{H}
 \end{equation}
 where $\Delta_i$ is the Laplacian with respect to $x_i$.
+(Throughout this document, we choose units so that $\hbar=1$; we keep track of the mass because different authors use $m=1$ or $m=1/2$, so we keep it to avoid confusion.)
 This is the microscopic description of the system.
 Similarly to the classical case, to treat situations with a huge number of particles, we will approach the system statistically.
 In the classical case, we considered a probability distribution over all possible configurations, without fixing the number of particles.
@@ -296,10 +297,9 @@ and the other is that exchanging two particles leads to the wavefunction changin
   .
 \end{equation}
 Why are these the only two possibilities (in three dimensions)?
-Mathematically, this comes from the fact that there are only two irreducible representations of the braid group in three dimensions.
+Mathematically, this comes from the fact that there are only two irreducible representations of the permutation group in three dimensions.
-More intuitively, if two particles are exchanged, the wavefunction should pick up a phase $e^{i\theta}$, but if I exchange the particles back, we should return to the original wavefunction, and thus pick up the phase $e^{-i\theta}$.
+More intuitively, if two particles are exchanged, the wavefunction should pick up a phase $e^{i\theta}$, but if I exchange the particles back, we should return to the original wavefunction, and thus $e^{2i\theta}=1$ and so $\theta$ is either $0$ or $\pi$.
-In three dimensions, there is no way of knowing which way is ``back'', so $e^{i\theta}=e^{-i\theta}$ and so $\theta$ is either $0$ or $\pi$.
+(This is not the case in two dimensions, for reasons we will not elaborate here.)
-(In two dimensions, one can know which way is ``back'', so there are other possible phases, but this is a discussion for another time).
 \bigskip
 
 \indent
@@ -636,6 +636,7 @@ Let us consider a more realistic Hamiltonian:
   \label{Ham}
 \end{equation}
 where $v$ is the potential that accounts for the interaction between pairs of particles.
+(Throughout this document, we choose units so that $\hbar=1$; we keep track of the mass because different authors use $m=1$ or $m=1/2$, so we keep it to avoid confusion.)
 This potential could be the Coulomb potential $1/|x|$ for charged particles, or the Lennard-Jones potential for interacting atoms, or something more realistic that accounts for the fine structure of atoms.
 Here, we will restrict our focus a bit and assume that $v$ is integrable ($v\in L_1(\mathbb R^3)$) and non-negative: $v(x)\geqslant 0$.
 In addition, we will only consider the zero-temperature state ($T=0$ that is $\beta=\infty$), in other words, we will be considering only the ground state of $H_N$, which is the eigenstate with lowest eigenvalue.
@@ -689,9 +690,10 @@ The physical picture to have in mind is two particles coming closer together in
 By translation invariance, we can work in the center of mass reference frame, in which the scattering event can be seen as a single particle flying through a fixed potential.
 The Schr\"odinger equation for this process is
 \begin{equation}
-  -\frac1m\Delta\psi+v(x)\psi=i\hbar\partial_t\psi
+  -\frac1m\Delta\psi+v(x)\psi=i\partial_t\psi
 \end{equation}
 (note the absence of a $1/2$ factor in front of the Laplacian which comes from the fact that the effective mass in the center of mass frame is $\frac{m_1m_2}{m_1+m_2}=m/2$.)
+(Recall that we chose units so that $\hbar=1$.)
 \bigskip
 
 \indent
@@ -814,11 +816,13 @@ Alternatively, the scattering length can be computed as follows.
 
 \theo{Lemma}\label{lemma:scattering}
   If $v$ is spherically symmetric, compactly supported and integrable, then
+  \nopagebreakaftereq
   \begin{equation}
     a=\frac m{4\pi}\int dx\ v(x)\psi(x)
     .
   \end{equation}
 \endtheo
+\restorepagebreakaftereq
 \bigskip
 
 \indent\underline{Proof}:
@@ -881,7 +885,7 @@ We will be interested in the {\it thermodynamic limit}, in which
   \frac NV=\rho
 \end{equation}
 with $\rho$ (the density) fixed.
-In this case, the Hamiltonian\-~(\ref{Ham}) has pure-point spectrum (it is a compact operator).
+In this case, the Hamiltonian\-~(\ref{Ham}) has pure-point spectrum (by Theorem\-~\ref{theo:compact_schrodinger}, $e^{-H_N}$ is compact, so it has discrete spectrum, and therefore so does $H_N$).
 Its lowest eigenvalue is denoted by $E_0$, and is called the {\it ground-state energy}.
 The corresponding eigenvector is denoted by $\psi_0$:
 \begin{equation}
@@ -908,11 +912,13 @@ Lee, Huang and Yang predicted the following low-density expansion for the ground
     \label{lhy_e}
   \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}
 \endtheo
+\restorepagebreakaftereq
 \bigskip
 
 The first two orders of this expansion thus only depend on the potential through its scattering length.
@@ -1314,6 +1320,7 @@ We thus define an equation obtained from the Complete Equation in which we drop
   \label{bigeqL}
   \end{equation}
 \endtheo
+\restorepagebreakaftereq
 \bigskip
 
 In practice, we have found that the predictions of the Big Equation are extremely close to those of the Complete Equation.
@@ -1711,7 +1718,7 @@ We are almost there: we have proved the existence of the solution of the modifie
 To prove Theorem\-~\ref{theo:existence}, we need to prove that $e\mapsto\rho(e)$ can be inverted locally.
 \bigskip
 
-\theo{Lemma}
+\theo{Lemma}\label{lemma:surjective}
   The map $e\mapsto\rho(e)$ is continuous, and $\rho(0)=0$ and $\lim_{e\to\infty}\rho(e)=\infty$.
   Therefore $e\mapsto\rho(r)$ can be inverted locally.
 \endtheo
@@ -1839,6 +1846,13 @@ But what of the uniqueness?
     .
   \end{equation}
 \qed
+\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.
 
 \subsection{Energy of the Simple Equation}
 \indent
@@ -2201,7 +2215,7 @@ In this appendix we gather a few useful definitions and results from functional
 \endtheo
 \bigskip
 
-\theoname{Theorem}{H\"older's inequality}\label{theo:holder}
+\theoname{Theorem}{H\"older's inequality \cite[Problem 0.26]{Te14}}\label{theo:holder}
   If $f\in L_p(\Omega)$ and $g\in L_q(\Omega)$, then $fg\in L_r(\Omega)$ with
   \begin{equation}
     \frac1r=\frac1p+\frac 1q
@@ -2253,42 +2267,78 @@ In this appendix we gather a few useful definitions and results from functional
 \bigskip
 
 \subsection{The Trotter product formula}
-\theoname{Theorem}{Trotter product formula}\label{theo:trotter}
+\theoname{Theorem}{Trotter product formula \cite[Theorem 5.11]{Te14}}\label{theo:trotter}
-  Given two operators $A$ and $B$ on a Banach space,
+  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^{A+B}
+    e^{-t(A+B)}
     =
-    \lim_{N\to\infty}\left(e^{\frac 1N A}e^{\frac1N B}\right)^N
+    \lim_{N\to\infty}\left(e^{-t\frac 1N A}e^{-t\frac1N B}\right)^N
     .
   \end{equation}
 \endtheo
 \restorepagebreakaftereq
 
+\subsection{Compact and trace-class operators}\label{app:compact}
+\theoname{Definition}{Trace}
+  Given a separable Hilbert space with an orthonormal basis $\{\varphi_i\}_{i\in\mathbb N}$, the trace of an operator $A$ is formally defined as
+  \begin{equation}
+    \mathrm{Tr}(A):=\sum_{i=0}^\infty \left<\varphi_i\right|A\left|\varphi_i\right>
+    .
+  \end{equation}
+  If this expression is finite, then $A$ is said to be trace-class.
+\endtheo
+\bigskip
+
+\theoname{Definition}{Compact operators}
+  The set of compact operators on a Hilbert space is the closure of the set of finite-rank operators.
+\endtheo
+\bigskip
+
+\theoname{Theorem}{\cite[Theorem 6.6]{Te14}}
+  If $A$ is a compact self-adjoint operator on a Hilbert space, then its spectrum consists of an at most countable set of eigenvalues.
+\endtheo
+\bigskip
+
+\theo{Theorem}\label{theo:trace_compact}
+  Trace class operators are compact.
+\endtheo
+\bigskip
+
+\theoname{Theorem}{\cite[Lemma 5.5]{Te14}}\label{theo:compact_ideal}
+  If $A$ is compact and $B$ is bounded, then $KA$ and $AK$ are compact.
+\endtheo
+\bigskip
+
+\theo{Theorem}\label{theo:compact_schrodinger}
+  Let $\Delta$ be the Laplacian on $L_2(\mathbb R^d/(L\mathbb Z)^d)$ (the same result holds for Dirichlet and Neumann boundary conditions as well).
+  For any function $v$ on $[0,L]^d$ that is bounded below, $e^{t(-\Delta+v)}$ is compact for any $t\geqslant 0$.
+\endtheo
+\bigskip
+
+\indent\underline{Proof}:
+  First of all, $e^{-t\Delta}$ is trace class: using the basis $e^{ikx}$ with $k\in(\frac{2\pi}L\mathbb Z)^d$
+  \begin{equation}
+    \mathrm{Tr}e^{-t\Delta}
+    =\sum_{k\in(\frac{2\pi}L\mathbb Z)^d}e^{-tk^2}<\infty
+    .
+  \end{equation}
+  By Theorem\-~\ref{theo:trace_compact}, $e^{-t\Delta}$ is compact.
+  By the Trotter product formula, Theorem\-~\ref{theo:trotter},
+  \begin{equation}
+    e^{-t\Delta+tv}
+    =\lim_{N\to\infty}(e^{-\frac tN\Delta}e^{\frac tNv})^N
+  \end{equation}
+  and by Theorem\-~\ref{theo:compact_ideal}, $(e^{-\frac tN\Delta}e^{\frac tNv})^N$ is compact.
+  Since the set of compact operators is closed, so is $e^{t(-\Delta+v)}$.
+\qed
+
 \subsection{Positivity preserving operators}\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}
-%  If $A$ and $B$ are positivity preserving, $A$ and $A^{-1}-B$ are invertible, and $\|BA\|<1$ or $\|AB\|<1$, then $(A^{-1}-B)^{-1}$ is positivity preserving.
-%\endtheo
-%\bigskip
-%
-%\indent\underline{Proof}:
-%  We expand
-%  \begin{equation}
-%    (A^{-1}-B)^{-1}
-%    =
-%    \sum_{n=0}^\infty A(BA)^n
-%    =
-%    \sum_{n=0}^\infty (AB)^nA
-%    .
-%  \end{equation}
-%\qed
-%\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.
 \endtheo
@@ -2321,20 +2371,6 @@ In this appendix we gather a few useful definitions and results from functional
 \qed
 \bigskip
 
-%\theo{Lemma}\label{lemma:diff}
-%  If $B$ is positivity preserving and $(A+\eta B)^{-1}$ is positivity preserving for all $\eta$, then for any $f\in\mathcal B_1$ such that $f\geqslant 0$, $\eta\mapsto(A+\eta B)^{-1}f$ is monotone decreasing pointwise in $x$.
-%\endtheo
-%\bigskip
-%
-%\indent\underline{Proof}:
-%  We have
-%  \begin{equation}
-%    \partial_\eta(A+\eta B)^{-1}f
-%    =-(A+\eta B)^{-1}B(A+\eta B)^{-1}f\leqslant 0
-%    .
-%  \end{equation}
-%\qed
-
 \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.
 \endtheo
@@ -2399,30 +2435,6 @@ In this appendix we gather a few useful definitions and results from functional
 {\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:yukawa}
-%  Given $\epsilon>0$, the operator $(-\Delta+\epsilon)^{-1}$ from $L_p(\mathbb R^3)$ to $W_{2,p}(\mathbb R^3)$ is positivity preserving.
-%\endtheo
-%\bigskip
-%
-%\indent\underline{Proof}:
-%  We apply lemma\-~\ref{lemma:add_inv} with $A^{-1}=-\Delta$
-%  We have
-%  \begin{equation}
-%    (-\Delta+\epsilon)^{-1}f(x)=\frac1{4\pi}\int dx\ \frac{e^{-\sqrt\epsilon|x-y|}}{|x-y|}f(y)
-%    .
-%  \end{equation}
-%\qed
-%\bigskip
-%
-%{\bf Example}:
-%This implies that if $v(x)\geqslant 0$, $v\in L_p(\mathbb R^3)$ and $\|v\|_p<\epsilon$, then $(-\Delta+\epsilon-v)^{-1}$ is positivity preserving.
-%Indeed, take $A=(-\Delta+\epsilon)^{-1}$, and use the fact that
-%\begin{equation}
-%  \|(-\Delta+\epsilon)^{-1}\|=\frac1\epsilon
-%  .
-%\end{equation}
-%\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
@@ -2475,7 +2487,7 @@ In this appendix, we state some useful results from harmonic analysis.
 \restorepagebreakaftereq
 \bigskip
 
-\theo{Theorem}\label{theo:harmonic}
+\theoname{Theorem}{\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
@@ -2499,12 +2511,14 @@ We need to compute these up to order $V^{-2}$, because one of the terms in the e
     u_3(x-y):=u(x-y)+\frac{w_3(x-y)}V
     \label{u3}
   \end{equation}
+  \nopagebreakaftereq
   \begin{equation}
     w_3(x-y):=(1-u(x-y))\int dz\ u(x-z)u(y-z)
     .
     \label{w3}
   \end{equation}
 \endtheo
+\restorepagebreakaftereq
 \bigskip
 
 \indent\underline{Proof}:
@@ -2900,6 +2914,26 @@ Check that Theorem\-~\ref{theo:schrodinger} applies, so that $e^{-tH_N}$ is posi
 Use the Perron-Frobenius theorem (Theorem\-~\ref{theo:perron_frobenius}) to conclude.
 \bigskip
 
+\problem\label{ex:nonneg} (solution on p.\-~\ref{sol:nonneg})\par
+\smallskip
+In this exercise, we will derive an alternate proof that $\psi_0\geqslant 0$ under the extra assumption that $v$ is continuous.
+To do so, consider the energy of $\psi_0$:
+\begin{equation}
+  \mathcal E(\psi_0):=\left<\psi_0\right|H_N\left|\psi_0\right>
+  =\int_{(\mathbb R/(L\mathbb Z))^{3N}} dx\ 
+  \left(
+    -\frac1{2m}\psi_0^*(x)\Delta\psi(x)
+    +V(x)|\psi_0(x)|^2
+  \right)
+\end{equation}
+where $\Delta$ is the Laplacian on $\mathbb R^{3N}$ and $V(x)\equiv\sum_{i<j}v(x_i-x_j)$.
+Prove that
+\begin{equation}
+  \mathcal E(|\psi_0|)=\mathcal E(\psi_0)
+\end{equation}
+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.
@@ -3078,10 +3112,47 @@ This implies\-~(\ref{sol_softcore}).
 
 \solution{perron_frobenius}
 By Theorem\-~\ref{theo:schrodinger}, $e^{-tH_N}$ is positivity preserving.
-In addition, $e^{-tH_N}$ is compact because $H_N$ is, and the spectrum of $e^{-tH_N}$ is $e^{-t\mathrm{spec}(H_N)}$, and the largest eigenvector of $e^{-tH_N}$ with the largest eigenvalue is the ground-state of $H_N$.
+In addition, $e^{-tH_N}$ is compact by Theorem\-~\ref{theo:compact_schrodinger}, and the spectrum of $e^{-tH_N}$ is $e^{-t\ \mathrm{spec}(H_N)}$, and the largest eigenvector of $e^{-tH_N}$ with the largest eigenvalue is the ground-state of $H_N$.
 Finally, since $\mathrm{spec}(H_N)\geqslant 0$ (because $-\Delta\geqslant 0$ and $v\geqslant 0$, we can apply the Perron-Frobenius theorem, which implies that $\psi_0$ is unique and $\geqslant 0$.
 \bigskip
 
+\solution{nonneg}
+The potential term $\int dx\ V(x)|\psi_0|^2$ is obviously the same for $\psi_0$ and $|\psi_0|$, so we only need to worry about the kinetic term.
+Let
+\begin{equation}
+  P:=\{x\in\mathbb R^{3N}:\ \psi_0(x)\neq0\}
+  .
+\end{equation}
+Now, $\psi_0$ is twice contiuously differentiable, since it is an eigenfunction and $v$ is continous (see\-~\cite[Theorem 11.7(vi)]{LL01}), so $|\psi_0|$ is twice continuously differentiable on $P$,  and so
+\begin{equation}
+  -\int dx\ |\psi_0|\Delta|\psi_0|
+  \equiv
+  -\int_P dx\ |\psi_0|\Delta|\psi_0|
+  =
+  \int_P dx\ (\nabla|\psi_0|)^2
+  +\int_{\partial P} dx\ |\psi_0(x)|(n(x)\cdot\nabla|\psi_0|)
+\end{equation}
+where $n(x)$ is the normal vector at $x$ ($\partial P$ is differentiable because $\psi_0$ is as well), but since $\psi_0(x)=0$ on $\partial P$,
+\begin{equation}
+  -\int dx\ |\psi_0|\Delta|\psi_0|
+  =
+  \int_P dx\ (\nabla|\psi_0|)^2
+  =
+  \int_P dx\ \left(\nabla\psi_0\frac{\psi_0}{|\psi_0|}\right)^2
+  =
+  \int_P dx\ (\nabla\psi_0)^2
+  =
+  -\int_P dx\ \psi_0\Delta\psi_0
+  .
+\end{equation}
+Thus, $\mathcal E(|\psi_0|)=\mathcal E(\psi_0)$.
+Since $\psi_0$ is the minimizer of $\mathcal E(\psi_0)$, so is $|\psi_0|$, but since the ground state is unique,
+\begin{equation}
+  \psi_0=|\psi_0|
+\end{equation}
+so $\psi_0\geqslant 0$.
+\bigskip
+
 \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

bibliography/bibliography.tex

@@ -29,7 +29,8 @@ doi:{\tt\color{blue}\href{http://dx.doi.org/10.4007/annals.2020.192.3.5}{10.4007
 \bibitem[FS23]{FS23}S. Fournais, J.P. Solovej - {\it The energy of dilute Bose gases II: the general case}, Inventiones mathematicae, volume\-~232, issue\-~2, pages\-~863-994, 2023,\par\penalty10000
 doi:{\tt\color{blue}\href{http://dx.doi.org/10.1007/s00222-022-01175-0}{10.1007/s00222-022-01175-0}}, arxiv:{\tt\color{blue}\href{http://arxiv.org/abs/2108.12022}{2108.12022}}.\par\medskip
  
-\bibitem[Ga99]{Ga99}G. Gallavotti - {\it Statistical mechanics, a short treatise}, Springer, 1999.\par\medskip
+\bibitem[Ga99]{Ga99}G. Gallavotti - {\it Statistical mechanics, a short treatise}, Springer, 1999,\par\penalty10000
+{\tt\color{blue}\href{https://141.108.10.74/pagine/deposito/1998/libro.pdf}{https://141.108.10.74/pagine/deposito/1998/libro.pdf}}.\par\medskip
  
 \bibitem[Ja22]{Ja22}I. Jauslin - {\it Review of a Simplified Approach to study the Bose gas at all densities}, The Physics and Mathematics of Elliott Lieb, The\-~90th Anniversary Volume I, chapter\-~25, pages\-~609-635, ed. Rupert L. Frank, Ari Laptev, Mathieu Lewin, Robert Seiringer, EMS Press, 2022,\par\penalty10000
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@@ -49,6 +50,8 @@ doi:{\tt\color{blue}\href{http://dx.doi.org/10.1103/PhysRev.130.2518}{10.1103/Ph
 \bibitem[LL64]{LL64}E.H. Lieb, W. Liniger - {\it Simplified Approach to the Ground-State Energy of an Imperfect Bose Gas. III. Application to the One-Dimensional Model}, Physical Review, volume\-~134, issue\-~2A, pages A312-A315, 1964,\par\penalty10000
 doi:{\tt\color{blue}\href{http://dx.doi.org/10.1103/PhysRev.134.A312}{10.1103/PhysRev.134.A312}}.\par\medskip
  
+\bibitem[LL01]{LL01}E.H. Lieb, M. Loss - {\it Analysis}, Second edition, Graduate studies in mathematics, Americal Mathematical Society, 2001.\par\medskip
+ 
 \bibitem[LS64]{LS64}E.H. Lieb, A.Y. Sakakura - {\it Simplified Approach to the Ground-State Energy of an Imperfect Bose Gas. II. Charged Bose Gas at High Density}, Physical Review, volume\-~133, issue\-~4A, pages A899-A906, 1964,\par\penalty10000
 doi:{\tt\color{blue}\href{http://dx.doi.org/10.1103/PhysRev.133.A899}{10.1103/PhysRev.133.A899}}.\par\medskip
  
@@ -66,6 +69,9 @@ 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
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+\bibitem[Te14]{Te14}G. 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
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