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\begin{document}
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\hbox{}\vfil
\bf\Large
\hfil Many interacting quantum particles:\par
\medskip
\hfil\large open problems, and a new point of view on an old problem
\vfil
\large
\hfil Ian Jauslin
\normalsize
\vfil
\rm
\hfil collaborators: {\bf E.A.\-~Carlen, E.H.\-~Lieb, M.\-~Holzmann, M.P.\-~Loss}\par
\vfil
arXiv:{\tt\ \parbox[b]{3cm}{
  \href{https://arxiv.org/abs/1912.04987}{1912.04987}\par
  \href{https://arxiv.org/abs/2010.13882}{2010.13882}\par
  \href{https://arxiv.org/abs/2011.10869}{2011.10869}
}}
\hfill{\tt \href{http://ian.jauslin.org}{http://ian.jauslin.org}}
\eject

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\title{Fermions/Bosons}
\bigskip
\hfil\includegraphics[height=5.5cm]{bosons-weinersmith.png}

\hfil{\tiny Zach Weinersmith, \href{https://creativecommons.org/licenses/by-nc/3.0/legalcode}{CC-BY-NC 3.0}}
\vfill
\eject

\title{Bose-Einstein condensation}
\begin{itemize}
\item System of Bosons, e.g. {\color{highlight}Helium} atoms, {\color{highlight}Rubidium} atoms, etc...
\item At low temperatures, {\color{highlight}superfluidity} (flow with zero viscocity) and {\color{highlight}superconductivity} (currents with zero resistance).
\item {\color{highlight}Bose-Einstein condensate}: most particles are in the same quantum state.
\item Predicted theoretically in {\color{highlight}1924-1925}, experimentally observed in {\color{highlight}1995}.
\item Mathematical understanding: still {\color{highlight}no proof} of the existence of a condensate (at finite density, in the presence of interactions and in the continuum).
\end{itemize}
\vfill
\eject

\title{Repulsive Bose gas}
\begin{itemize}
  \item {\color{highlight}$N$-particle} quantum state in a volume $V$:
  $$
    \psi(x_1,\cdots,x_N)\in L^2_{\mathrm{symmetric}}((V\mathbb T^3)^N)
  $$

  \item $|\psi|^2$: probability distribution on the positions of the $N$ particles.

  \item Hamiltonian operator acting on $\psi$:
  $$
    H_N\psi:=
    -\frac12\sum_{i=1}^N\Delta_i\psi
    +\sum_{1\leqslant i<j\leqslant N}{\color{highlight}v(|x_i-x_j|)}\psi
  $$
  {\color{highlight}$v(r)\geqslant 0$}.
\vfill
\eject

\title{Energy and condensate fraction}
\vskip-10pt
  \item {\color{highlight}Zero-temperature:} Ground state: $\psi_0$, energy $E_0=\inf\mathrm{spec}H_N$:
  $$
    H_N\psi_0={\color{highlight}E_0}\psi_0
  $$
\vskip-10pt
  \item In the {\color{highlight}thermodynamic limit}:
  $$
    e_0:=\lim_{\displaystyle\mathop{\scriptstyle V,N\to\infty}_{{\color{highlight}\frac NV=\rho}}}\frac{E_0}N
    .
  $$
\vskip-5pt
  \item Condensate fraction: proportion of particles in the Bose-Einstein condensate: in the {\color{highlight}thermodynamic limit}: ($P_i$: projection onto condensate wavefunction)
  $$
    \eta_0:=\lim_{\displaystyle\mathop{\scriptstyle V,N\to\infty}_{{\color{highlight}\frac NV=\rho}}}\frac1N\left<\psi_0\right|\sum_{i=1}^NP_i\left|\psi_0\right>
    .
  $$
\end{itemize}
\vfill
\eject

\title{Ground state energy}
\begin{itemize}
  \item At low density: {\color{highlight}Bogolyubov theory}: [Bogolyubov, 1947], \href{https://doi.org/10.1103/PhysRev.106.1135}{[Lee, Huang, Yang, 1957]}:
  $$
    e_0=2\pi\rho a\left(1+\frac{128}{15\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt\rho)\right)
  $$
  \item {\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/2101.06222}{[Basti, Cenatiempo, Schlein, 2021]}),
  \href{https://doi.org/10.4007/annals.2020.192.3.5}{[Fournais, Solovej, 2020]}.

  \item At high density: {\color{highlight}Hartree theory}: ({\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{Condensate fraction}
\begin{itemize}
  \item At low density: {\color{highlight}Bogolyubov theory}: [Bogolyubov, 1947], \href{https://doi.org/10.1103/PhysRev.106.1135}{[Lee, Huang, Yang, 1957]}:
  $$
    1-\eta_0\sim\frac{8\sqrt{\rho a^3}}{3\sqrt\pi}
  $$

  \item {\color{highlight}Still open} in the thermodynamic limit. (No proof of Bose-Einstein condensation, in the continuum, at finite density.)

  \item At high density: {\color{highlight}Hartree theory}: ({\color{highlight}open})
  $$
    \eta_0\to1
  $$
\end{itemize}
\vfill
\eject

\title{Effective equations}
\begin{itemize}
  \item {\color{highlight}Boltzmann equation}: $N$ classical hard particles with an infinitely small radius (dilute limit)
  [Lanford, 1976].
  \item {\color{highlight}Thomas-Fermi theory}: $Z$ electrons orbiting a nucleus in the $Z\to\infty$ limit
  \href{https://doi.org/10.1103/PhysRevLett.31.681}{[Lieb, Simon, 1973]}.
  \item{\color{highlight}Hartree-Fock equation}: dynamics of many Fermions in the weakly-interacting limit
  \href{https://doi.org/10.1142/9789814618144_0011}{[Benedikter, Porta, Schlein, 2015]}.
  \item{\color{highlight}Hartree-Fock-Bogolyubov equation}: dynamics of many Bosons in the weakly-interacting limit
  \href{https://arxiv.org/abs/1602.05171}{[Bach, Breteaux, Chen, Fr\"ohlich, Sigal, 2016]}.
\end{itemize}
\vfill
\eject

\title{{\color{iblue}Simple equation}}
\begin{itemize}
  \item {\color{iblue}Simple equation}
  $$
    -\Delta u(x)=(1-u(x))v(x)- 4eu(x)+2e\rho\ u\ast u(x)
  $$
  $$
    e=\frac\rho2\int dx\ (1-u(x))v(x)
  $$
  \item $\rho>0$, $v(x)\geqslant 0$, $v\in L_1(\mathbb R^3)$.
  \item {\color{highlight}Non-linear} and {\color{highlight}non-local} partial differential equation.
  \item {\color{highlight}Effective equation} for the ground state of a Bose gas.
  \item Main idea: think of {\color{highlight}$\psi$ as a probability distribution} instead of $|\psi|^2$.
\end{itemize}
\vfill
\eject

\title{Energy as a function of density for the {\color{iblue}Simple equation}}
For $v(x)=e^{-|x|}$:

\hfil\includegraphics[height=5.5cm]{erho_lowhigh.pdf}
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\title{Energy as a function of density for the {\color{iblue}Simple equation}}
For $v(x)=e^{-|x|}$:

\hfil\includegraphics[height=5.5cm]{erho_effective.pdf}
\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{{\color{ipurple}Big equation}}
\begin{itemize}
\item $x\in\mathbb R^3$,
  $$
    -\Delta u(x)
    =
    (1-u(x))\left(v(x)-2\rho K(x)+\rho^2 L(x)\right)
  $$
  $$
    K:=
    u\ast S
    ,\quad
    S(y):=(1-u(y))v(y)
  $$
  $$
    L:=
    u\ast u\ast S
    -2u\ast(u(u\ast S))
    .
  $$
\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{Conclusions}
\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 Bogolyubov 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}

\end{document}