Update to v0.1:

Fixed: H_N is not compact, e^{-H_N} is.

  Fixed: Removed confusing discussion of Anyons

  Added: Discussion of compactness.

Changelog

@@ -0,0 +1,8 @@
+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

@@ -296,10 +296,9 @@ 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.
-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 $e^{-i\theta}$.
-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 $\pi$.
-(In two dimensions, one can know which way is ``back'', so there are other possible phases, but this is a discussion for another time).
+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 $e^{2i\theta}=1$ and so $\theta$ is either $0$ or $\pi$.
+(This is not the case in two dimensions, for reasons we will not elaborate here.)
 \bigskip
 
 \indent
@@ -881,7 +880,7 @@ 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}
@@ -2201,7 +2200,7 @@ 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
@@ -2253,18 +2252,72 @@ 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}
-  Given two operators $A$ and $B$ on a Banach space,
+\theoname{Theorem}{Trotter product formula \cite[Theorem 5.11]{Te14}}\label{theo:trotter}
+  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 $f(x)\geqslant0$, $Af(x)\geqslant 0$.
@@ -2475,7 +2528,7 @@ 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
@@ -3078,7 +3131,7 @@ 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
 

bibliography/bibliography.tex

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+\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
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@@ -66,6 +68,8 @@ 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\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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