\documentclass{ian-presentation}

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\definecolor{ipurple}{HTML}{4B0082}
\definecolor{iyellow}{HTML}{DAA520}
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\definecolor{highlight}{HTML}{328932}
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\begin{document}
\pagestyle{empty}
\hbox{}\vfil
\bf\Large
\hfil An effective equation to study Bose gasses\par
\smallskip
\hfil at all densities\par
\vfil
\large
\hfil Ian Jauslin
\normalsize
\vfil
\hfil\rm joint with {\bf Eric A. Carlen}, {\bf Markus Holzmann}, {\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

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\title{Repulsive Bose gas}
\begin{itemize}
  \item Potential: {\color{highlight}$v(r)\geqslant 0$} and {\color{highlight}$v\in L_1(\mathbb R^3)$}, Hamiltonian:
  $$
    H_N:=
    -\frac12\sum_{i=1}^N\Delta_i
    +\sum_{1\leqslant i<j\leqslant N}v(|x_i-x_j|)
  $$
  \item Ground state: $\psi_0$, energy $E_0$.
  \item Observables in the {\color{highlight}thermodynamic limit}: ground state energy per particle and condensate fraction:
  $$
    e_0:=\lim_{\displaystyle\mathop{\scriptstyle V,N\to\infty}_{{\color{highlight}\frac NV=\rho}}}\frac{E_0}N
    ,\quad
    \eta_0:=\lim_{\displaystyle\mathop{\scriptstyle V,N\to\infty}_{{\color{highlight}\frac NV=\rho}}}\frac1N\left<\psi_0\right|\sum_{i=1}^N\int\frac{dx_i}V\left|\psi_0\right>
    .
  $$
\end{itemize}
\vfill
\eject

\title{Effective theories}
\begin{itemize}
  \item {\color{highlight}Bogolubov} theory: {\color{highlight}approximation scheme} that makes $H$ {\color{highlight}integrable}.
  \item Low density predictions \href{https://doi.org/10.1103/PhysRev.106.1135}{[Lee, Huang, Yang, 1957]}:
  \begin{itemize}
    \item Energy:
    $$
      {\color{highlight}e_0=2\pi\rho a\left(1+\frac{128}{15\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt\rho)\right)}
    $$
    \vskip-10pt
    \item Condensate fraction:
    $$
      {\color{highlight}1-\eta_0\sim\frac{8\sqrt{\rho a^3}}{3\sqrt\pi}}
    $$
  \end{itemize}
  \item At high density: {\color{highlight}Hartree mean-field theory}.
\end{itemize}
\vfill
\eject

\title{Mathematical results}
\begin{itemize}
  \item Energy asymptotics: {\color{highlight} proved}:
  \href{https://doi.org/10.1103/PhysRevLett.80.2504}{[Lieb, Yngvason, 1998]},
  \href{https://doi.org/10.1007/s10955-009-9792-3}{[Yau, Yin, 2009]},
  \href{https://arxiv.org/abs/1904.06164}{[Fournais, Solovej, 2019]}

  \item Condensate fraction: {\color{highlight}still open} in the thermodynamic limit.

  \item At high density, {\color{highlight}proved} in \href{https://doi.org/10.1103/PhysRev.130.2518}{[Lieb, 1963]}.
  $$
    {\color{highlight}e_0\sim\frac\rho2\int v}
  $$
\end{itemize}
\vfill
\eject

\title{Energy as a function of density}
For $v(x)=e^{-|x|}$:

\hfil\includegraphics[height=5.5cm]{erho_lowhigh.pdf}
\vfill
\eject

\addtocounter{page}{-1}
\title{Energy as a function of density}
For $v(x)=e^{-|x|}$:

\hfil\includegraphics[height=5.5cm]{erho_effective.pdf}
\vfill
\eject

\title{{\color{ipurple}Big equation}}
\begin{itemize}
\item Solve for
\begin{equation}
  u(x_2-x_1):=1-\frac{\int \frac{dx_3}V\cdots\frac{dx_N}V\ \psi_0(x_1,\cdots,x_N)}{\int \frac{dx_1}V\cdots\frac{dx_N}V\ \psi_0(x_1,\cdots,x_N)}
\end{equation}
\item {\color{ipurple}``Big'' equation}:
  $${\color{ipurple}
    -\Delta u(x)
    =
    (1-u(x))\left(v(x)-2\rho u\ast S(x)+\rho^2 u\ast u\ast S(x)\right)
  }$$
  $$
    S(y):=(1-u(y))v(y)
  $$
\end{itemize}
\vfill
\eject
  
\title{{\color{iblue}Simple equation}}
\vskip-10pt
\begin{itemize}
  \item Further approximate $S(x)\approx\frac{2e}\rho\delta(x)$ and $u\ll 1$.
  \item {\color{iblue}Simple equation}:
  $$
    {\color{iblue}-\Delta u(x)=(1-u(x))v(x)- 4eu(x)+2e\rho\ u\ast u(x)}
  $$
  $$
    e=\frac\rho2\int dx\ (1-u(x))v(x)
  $$
  \item {\bf Theorem 1}:
  If $v(x)\geqslant 0$ and $v\in L_1\cap L_2(\mathbb R^3)$, then the {\color{iblue}simple equation} has an {\color{highlight}integrable solution} (proved constructively), with $0\leqslant u\leqslant 1$.
\end{itemize}
\vfill
\eject

\title{Energy for the {\color{iblue}simple equation}}
\vskip-10pt
\begin{itemize}
  \item {\bf Theorem 2}:
  $$
    \frac{e}{\rho}\mathop{\longrightarrow}_{\rho\to\infty}\frac12\int dx\ v(x)
  $$
  (note that there is no condition that $\hat v\geqslant 0$).
  This coincides with the {\color{highlight}Hartree energy}.
  \item {\bf Theorem 3}:
  $$
    e=2\pi\rho a\left(1+\frac{128}{15\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt\rho)\right)
  $$
  This coincides with the {\color{highlight}Lee-Huang-Yang prediction}.
\end{itemize}
\vfill
\eject

\title{Energy}
$v(x)=e^{-|x|}$, Blue: {\color{iblue}simple equation}; purple: {\color{ipurple}big equation}; red: {\color{ired}Monte Carlo}

\hfil\includegraphics[height=5.5cm]{erho_fulleq.pdf}
\vfill
\eject

\title{Condensate fraction}
\begin{itemize}
  \item {\bf Theorem 4}:
  For the {\color{iblue}simple equation}, as $\rho\to0$
  $$
    1-\eta\sim\frac{8\sqrt{\rho a^3}}{3\sqrt\pi}
  $$
  which coincides with {\color{highlight}Bogolubov's prediction}.
\end{itemize}
\vfill
\eject

\title{Condensate fraction}
$v(x)=e^{-|x|}$, Blue: {\color{iblue}simple equation}; purple: {\color{ipurple}big equation}; red: {\color{ired}Monte Carlo}

\hfil\includegraphics[height=5.5cm]{condensate.pdf}
\vfill
\eject

\title{Limitations of the simple and big equations}
\begin{itemize}
  \item Only works at high densities for {\color{highlight}$\hat v\geqslant 0$}.
  \item Less accurate for {\color{highlight}large potentials}: for $v(x)=16e^{-|x|}$,

  \hfil\includegraphics[width=5.5cm]{energy16.pdf}
  \hfil\includegraphics[width=5.5cm]{condensate16.pdf}
\end{itemize}
\vfill
\eject

\title{Conclusions and outlooks}
\begin{itemize}
  \item Two {\color{highlight}effective equations}: the {\color{ipurple}big equation} and the {\color{iblue}simple equation}, which are {\color{highlight}non-linear 1-particle equations}.
  \item Reproduce the known results for both {\color{highlight}small and large densities}.
  \item Their derivation is {\color{highlight}different from Bogolubov theory}, so they may give new insights onto studying the Bose gas in these asymptotic regimes.
  \item The {\color{ipurple}big equation} is {\color{highlight}quantitatively accurate} at intermediate densities.
  \item This opens up the possibility of studying the physics of the {\color{highlight}Bose gas at intermediate densities}.
\end{itemize}
\vfill
\eject

\title{Open problems}
\begin{itemize}
  \item Analysis of the {\color{iblue}simple equation}: {\color{highlight}Monotonicity} of $e(\rho)$, and {\color{highlight}convexity} of $\rho e(\rho)$. (So far, we have proofs for small and large $\rho$.). Similarly, prove that $0\leqslant\eta\leqslant 1$. (We have a proof for small $\rho$.)

  \item Analysis of the {\color{ipurple}big equation}: everything is still open.

  \item Relate these equations back to the {\color{highlight}many-body Bose gas}.
\end{itemize}

\end{document}