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\documentclass{ian-presentation}
\usepackage[hidelinks]{hyperref}
\usepackage{graphicx}
\usepackage{array}
\begin{document}
\pagestyle{empty}
\hbox{}\vfil
\bf\Large
\hfil Lieb's simplified approach to interacting Bose gases\par
\vfil
\large
\hfil Ian Jauslin
\normalsize
\vfil
\hfil\rm joint with {\bf Eric Carlen}, {\bf Elliott H. Lieb}\par
\vfil
\hfill{\tt \href{http://ian.jauslin.org}{http://ian.jauslin.org}}
\eject
\setcounter{page}1
\pagestyle{plain}
\title{Bose gas}
\hfil\includegraphics[height=6cm]{BEC.png}
\vfill
\eject
\title{Interacting Bose gas}
\begin{itemize}
\item State: symmetric wave functions in a finite box of volume $V$ with periodic boundary conditions:
$$
\psi(x_1,\cdots,x_N)
,\quad
x_i\in \Lambda_d:=V^{\frac1d}\mathbb T^d
$$
\item $N$-particle Hamiltonian:
$$
H_N:=
-\frac12\sum_{i=1}^N\Delta_i
+\sum_{1\leqslant i<j\leqslant N}v(|x_i-x_j|)
$$
with a repulsive, integrable interaction: $v(|x-y|)\geqslant 0$, $\int dx\ v(x)<\infty$.
\end{itemize}
\vfill
\eject
\title{Interacting Bose gas}
\begin{itemize}
\item Ground state:
$$
H_N\psi_0=E_0\psi_0
,\quad
E_0=\min\mathrm{spec}(H_N)
$$
\item Compute the ground state-energy per particle in the thermodynamic limit:
$$
e_0:=\lim_{\displaystyle\mathop{\scriptstyle V,N\to\infty}_{\frac NV=\rho}}\frac{E_0}N
$$
\end{itemize}
\vfill
\eject
\title{Energy}
\begin{itemize}
\item Integrate $H_N\psi_0=E_0\psi_0$:
$$
\frac{E_0}N=\frac{N-1}{2V}\int dx\ v(|x-y|)g_2(x,y)
$$
\item $g_n$: marginal of $\psi_0$
$$
g_n(x_1,\cdots,x_n):=\frac{V^n\int dx_{n+1}\cdots dx_N\ \psi_0(x_1,\cdots,x_N)}{\int dx_1\cdots dx_N\ \psi_0(x_1,\cdots,x_N)}
$$
\item $\psi_0\geqslant 0$, so it can be thought of as a probability distribution
\end{itemize}
\vfill
\eject
\title{Hierarchy}
\vskip-10pt
\begin{itemize}
\item Equation for $g_2$: integrate $H_N\psi_0=E_0\psi_0$ with respect to $x_3,\cdots,x_N$:
$$
\begin{array}{>\displaystyle l}
-\frac12(\Delta_x+\Delta_y) g_2(x,y)
+\frac{N-2}V\int dz\ (v(|x-z|)+v(|y-z|))g_3(x,y,z)
\\[0.5cm]\hfill
+v(|x-y|)g_2(x,y)
+\frac{(N-2)(N-3)}{2V^2}\int dzdt\ v(|z-t|)g_4(x,y,z,t)
=E_0g_2(x,y)
\end{array}
$$
\item Factorization assumption:
$$
g_3(x_1,x_2,x_3)=g_2(x_1,x_2)g_2(x_1,x_3)g_2(x_2,x_3)
$$
$$
g_4(x_1,x_2,x_3,x_4)=\prod_{i<j}(g_2(x_i,x_j)+O(V^{-1}))
$$
\end{itemize}
\vfill
\eject
\title{Lieb's simple equation}
\begin{itemize}
\item In the thermodynamic limit, after making a few additional assumptions, \href{https://doi.org/10.1103/PhysRev.130.2518}{[Lieb, 1963]}:
$$
(-\Delta+v(|x|)+4e_0)u(|x|)=v(|x|)+2e_0\rho\ u\ast u(|x|)
$$
$$
e_0=\frac\rho2\int dx\ (1-u(|x|))v(|x|)
$$
\item with $\rho:=\frac NV$
$$
g_2(x,y)=1-u(|x-y|)
,\quad
u\ast u(|x|):=\int dy\ u(|x-y|)u(|y|)
$$
\end{itemize}
\vfill
\eject
\title{One dimension}
\hfil\includegraphics[height=6cm,angle=0.4]{1d.png}
\vfill
\eject
\title{Numerical solution for $v(x)=e^{-|x|}$ in 3 dimensions}
\hfil\includegraphics[height=6cm]{erho_bare.pdf}
\vfill
\eject
\title{Numerical solution for $v(x)=e^{-|x|}$ in 3 dimensions}
\hfil\includegraphics[height=6cm]{erho.pdf}
\vfill
\eject
\title{Asymptotics}
\vskip-10pt
\begin{itemize}
\item {\bf Theorem} \href{https://doi.org/10.1103/PhysRev.130.2518}{[Lieb, 1963]}: if $\hat v(k):=\int dx\ e^{ikx}v(x)\geqslant 0$, then
$$
\frac{e_0}\rho\mathop{\longrightarrow}_{\rho\to\infty}\frac12\int dx\ v(x)
$$
\item {\bf Theorem} \href{https://doi.org/10.1103/PhysRevLett.80.2504}{[Lieb, Yngvason, 1998]}: in 3 dimensions
$$
\frac{e_0}\rho\mathop{\longrightarrow}_{\rho\to0}2\pi a
$$
\item {\bf Theorem} \href{https://doi.org/10.1023/A:101033721}{[Lieb, Yngvason, 2001]}: in 2 dimensions
$$
\frac{e_0}\rho\mathop{\longrightarrow}_{\rho\to0}\frac{2\pi}{|\log(\rho a^2)|}
$$
\end{itemize}
\vfill
\eject
\title{Lee-Huang-Yang formula}
\begin{itemize}
\item In 3 dimensions, \href{https://doi.org/10.1103/PhysRev.106.1135}{[Lee, Huang, Yang, 1957]} conjecture:
$$
e_0=2\pi\rho a\left(1+\frac{128}{15\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt\rho)\right)
$$
\item This was recently proved by \href{https://arxiv.org/abs/1904.06164}{[Fournais, Solovej, 2019]}.
\end{itemize}
\vfill
\eject
\title{Results}
\begin{itemize}
\item {\bf Theorem}: For Lieb's simplified equation,
$$
\frac{e_0}\rho\mathop{\longrightarrow}_{\rho\to\infty}\frac12\int dx\ v(x)
$$
and, in 3 dimensions,
$$
e_0=2\pi\rho a\left(1+\frac{128}{15\sqrt\pi}\sqrt{\rho a^3}+O(\rho)\right)
$$
in 2 dimensions
$$
\frac{e_0}\rho\mathop{\longrightarrow}_{\rho\to\infty}\frac{2\pi}{|\log(\rho a)|}
.
$$
\end{itemize}
\end{document}
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