\documentclass{ian-presentation}

\usepackage[hidelinks]{hyperref}
\usepackage{graphicx}
\usepackage{array}

\begin{document}
\pagestyle{empty}
\hbox{}\vfil
\bf\Large
\hfil A 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
arXiv:{\tt \href{https://arxiv.org/abs/1912.04987}{1912.04987}}
\hfill{\tt \href{http://ian.jauslin.org}{http://ian.jauslin.org}}
\eject

\setcounter{page}1
\pagestyle{plain}
\title{Lieb's simple equation}
\vskip-10pt
\begin{itemize}
  \item \href{https://doi.org/10.1103/PhysRev.130.2518}{[Lieb, 1963]}: $x\in\mathbb R^3$
  $$
    (-\Delta+v(x)+4e)u(x)=v(x)+2e\rho\ u\ast u(x)
  $$
  $$
    e=\frac\rho2\int dx\ (1-u(x))v(x)
  $$
  \item with
  $$
    \rho>0
    ,\quad
    v(x)\geqslant 0
    ,\quad
    v\in L_1\cap L_2(\mathbb R^3)
  $$
  \item and
  $$
    u(x)\geqslant 0
    ,\quad
    u\in L_1(\mathbb R^3)
    ,\quad
    u\ast u(x):=\int dy\ u(x-y)u(y)
  $$
\end{itemize}
\vfill
\eject


\title{Interacting Bose gas}
\vskip-10pt
\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 Probability distribution: $|\psi(x_1,\cdots,x_N)|^2$
  \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 $v(x-y)\geqslant 0$ and $v\in L_1\cap L_{\frac d2+\epsilon}(\mathbb R^d)$.
\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 $H_N\psi_0=E_0\psi_0$:
  $$
    \frac{E_0}N=\frac{N-1}{2V}\int dx\ v(x)g_2(0,x)
  $$
  \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]}:
  $$
    (-\Delta+v(x)+4e)u(x)=v(x)+2e\rho\ u\ast u(x)
  $$
  $$
    e=\frac\rho2\int dx\ (1-u(x))v(x)
  $$
  \item with $\rho:=\frac NV$
  $$
    g_2(x,y)=1-u(x-y)
  $$
\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 for the Bose gas}
\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 ($a$: scattering length)
  $$
    \frac{e_0}{\rho}\mathop{\longrightarrow}_{\rho\to0}2\pi a
  $$
  \href{https://doi.org/10.1103/PhysRev.106.1135}{[Lee, Huang, Yang, 1957]}, \href{https://doi.org/10.1007/s10955-009-9792-3}{[Yau, Yin, 2009]}, \href{https://arxiv.org/abs/1904.06164}{[Fournais, Solovej, 2019]}:
  $$
    e_0=2\pi\rho a\left(1+\frac{128}{15\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt\rho)\right)
  $$
\end{itemize}
\vfill
\eject

\title{Comparison with Bose gas (Monte Carlo)}
\vskip-10pt
\hfil\includegraphics[width=8cm]{erho_holzmann.pdf}

\hfil{\footnotesize Monte Carlo computation courtesy of M. Holzmann}
\vfill
\eject

\title{Main Theorem}
\vskip-5pt
\begin{itemize}
  \item If $v(x)\geqslant 0$ and $v\in L_1\cap L_{\frac d2+\epsilon}(\mathbb R^d)$, then Lieb's simple equation
  $$
    (-\Delta+4e+v)u=v+2e\rho u\ast u
    ,\quad
    e=\frac\rho2\int dx\ (1-u(x))v(x)
  $$
  has an integrable, non-negative solution (proved constructively).

  \item In 3 dimensions,
  $$
    e=2\pi\rho a\left(1+\frac{128}{15\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt\rho)\right)
    ,\quad
    \frac{e}\rho\mathop{\longrightarrow}_{\rho\to\infty}\frac12\int dx\ v(x)
  $$

  \item If $v(x)\equiv v(|x|)$ is radially symmetric and decays exponentially,
  $$
    u(|x|)\mathop\sim_{|x|\to\infty}\frac\alpha{|x|^4}
  $$
\end{itemize}
\vfill
\eject

\title{Existence of a solution (sketch)}
\begin{itemize}
  \item Change the point of view: fix $e>0$, and compute $\rho$ and $u$.

  \item Iteration: $u_0=0$,
  $$
    (-\Delta+4e+v)u_n=v+2e\rho_{n-1}u_{n-1}\ast u_{n-1}
    ,\quad
    \rho_n:=\frac{2e}{\int dx\ (1-u_n(x))v(x)}
    .
  $$

  \item Prove by induction that $u_n(x)$ is an increasing sequence, and is bounded $u_n(x)\leqslant 1$. It therefore converges to a function $u$, which is the unique integrable non-negative solution of the equation with $e$ fixed.

  \item In addition, we prove that $e\mapsto\rho(e)$ is continuous, and $\rho(0)=0$ and $\rho(\infty)=\infty$, which allows us to  compute solutions for the problem at fixed $\rho$. This does not imply the uniqueness of the solution.
\end{itemize}
\vfill
\eject

\title{Asymptotics (sketch)}
\vskip-10pt
\begin{itemize}
  \item When $\rho$ is small, $e$ is small as well, so the solution $u$ is {\it not too far from} the solution of the scattering equation
  $$
    (-\Delta+v)\varphi=v
    .
  $$

  \item The energy of $\varphi$ is
  $$
    \frac\rho 2\int dx\ (1-\varphi(x))v(x)=2\pi\rho a
  $$
  which yields the first term in the expansion.

  \item The second term comes from approximating
  $$
    (1-u(x))v(x)\approx\frac{2e}\rho\delta(x)
  $$
  and solving the equation in Fourier space.
\end{itemize}
\vfill
\eject

\title{Decay (sketch)}
$$
  (-\Delta+4e+v)u=v+2e\rho u\ast u
  ,\quad
  e=\frac\rho2\int dx\ (1-u(x))v(x)
$$
\begin{itemize}
  \item $u$ and $u\ast u$ have to decay at the same rate. This is a property of algebraically decaying functions.

  \item (Remark: $u_n(x)$ decays exponentially).

  \item Proof is based on the Fourier transform and complex analysis.

  \item Remark: The truncated two-point correlation function of the Bose gas is also conjectured to decay like $|x|^{-4}$.
\end{itemize}
\vfill
\eject

\title{Conclusion}
\vfill
\begin{itemize}
  \item Simple equation: correct asymptotics for the ground state energy at both high and low densities.

  \item Good approximation for intermediate densities (relative error of 5\%).

  \item Intriguing non-linear PDE.

  \item Proved existence, asymptotics, and decay rate.
\end{itemize}
\vfill
\eject

\title{Open problems and conjectures}
\begin{itemize}
  \item Monotonicity of $e\mapsto\rho(e)$, and concavity of $e\mapsto\frac1{\rho(e)}$ (would imply uniqueness).

  \item Existence of solutions that are not non-negative? (seems unlikely)

  \item Other observables? Condensate fraction? (in progress)

  \item Crystallization?

  \item {\it Lieb's simple equation} is actually a simplified version of a more complicated one: {\it Lieb's full equation}. Can it improve on the simple one? (in progress)
\end{itemize}

\end{document}
