arXiv subjects
Publications and source records attributed to Benjamin Schlein.
We consider systems of $N$ bosons in $\mathbb{R}^3$, trapped by an external potential. The interaction is repulsive and has a scattering length of the order $N^{-1}$ (Gross-Pitaevskii regime). We determine the ground state energy and the low-energy excitation spectrum up to errors that vanish in the limit $N\to \infty$.
We prove an adiabatic theorem for the Landau-Pekar equations. This allows us to derive new results on the accuracy of their use as effective equations for the time evolution generated by the Fröhlich Hamiltonian with large coupling constant $α$. In particular, we show that the time evolution of Pekar product states with coherent phonon field and the electron being trapped by the phonons is well approximated by the Landau-Pekar equations until times short compared to $α^2$.
We consider the quantum many-body evolution of a homogeneous Fermi gas in three dimensions in the coupled semiclassical and mean-field scaling regime. We study a class of initial data describing collective particle-hole pair excitations on the Fermi ball. Using a rigorous version of approximate bosonization, we prove that the many-body evolution can be approximated in Fock space norm by a quasifree bosonic evolution of the collective particle-hole excitations.
We consider a Bose gas consisting of $N$ particles in $\mathbb{R}^3$, trapped by an external field and interacting through a two-body potential with scattering length of order $N^{-1}$. We prove that low energy states exhibit complete Bose-Einstein condensation with optimal rate, generalizing previous work in \cite{BBCS1, BBCS4}, restricted to translation invariant systems. This extends recent results in \cite{NNRT}, removing the smallness assumption on the size of the scattering length.
We prove that the grand canonical Gibbs state of an interacting quantum Bose gas converges to the Gibbs measure of a nonlinear Schrödinger equation in the mean-field limit, where the density of the gas becomes large and the interaction strength is proportional to the inverse density. Our results hold in dimensions $d \leq 3$. For $d > 1$ the Gibbs measure is supported on distributions of negative regularity and we have to renormalize the interaction. More precisely, we prove the convergence of the relative partition function and of the reduced density matrices in the $L^r$-norm with optimal exponent $r$. Moreover, we prove the convergence in the $L^\infty$-norm of Wick-ordered reduced density matrices, which allows us to control correlations of Wick-ordered particle densities as well as the asymptotic distribution of the particle number. Our proof is based on a functional integral representation of the grand canonical Gibbs state, in which convergence to the mean-field limit follows formally from an infinite-dimensional stationary phase argument for ill-defined non-Gaussian measures. We make this argument rigorous by introducing a white-noise-type auxiliary field, through which the functional integral is expressed in terms of propagators of heat equations driven by time-dependent periodic random potentials and can, in turn, be expressed as a gas of interacting Brownian loops and paths. When the gas is confined by an external trapping potential, we control the decay of the reduced density matrices using excursion probabilities of Brownian bridges.
We consider Bose gases of $N$ particles in a box of volume one, interacting through a repulsive potential with scattering length of order $N^{1-κ}$, for $κ> 0$. Such regimes interpolate between the Gross-Pitaevskii and thermodynamic limits. Assuming that $κ$ is sufficiently small, we determine the ground state energy and the low-energy excitation spectrum of the system, up to errors vanishing in the limit of large $N$.
We consider a gas of $N$ bosons in a box with volume one interacting through a two-body potential with scattering length of order $N^{-1}$ (Gross-Pitaevskii limit). Assuming the (unscaled) potential to be sufficiently small, we show that the ground state of the system and all states with relatively small excitation energy exhibit complete Bose-Einstein condensation, with a uniform (i.e. $N$ independent) bound on the number of excitations.
We derive rigorously the leading order of the correlation energy of a Fermi gas in a scaling regime of high density and weak interaction. The result verifies the prediction of the random-phase approximation. Our proof refines the method of collective bosonization in three dimensions. We approximately diagonalize an effective Hamiltonian describing approximately bosonic collective excitations around the Hartree-Fock state, while showing that gapless and non-collective excitations have only a negligible effect on the ground state energy.
We establish an upper bound for the ground state energy per unit volume of a dilute Bose gas in the thermodynamic limit, capturing the correct second order term, as predicted by the Lee-Huang-Yang formula. This result has been first established in J. Stat. Phys. 136(3) by H.-T. Yau and J. Yin. Our proof, which applies to repulsive and compactly supported $V \in L^3 (\mathbb{R}^3)$, gives better rates and, in our opinion, is substantially simpler.
We consider the many-body quantum evolution of a factorized initial data, in the mean-field regime. We show that fluctuations around the limiting Hartree dynamics satisfy large deviation estimates, that are consistent with central limit theorems that have been established in the last years.
We consider the Fröhlich Hamiltonian with large coupling constant $α$. For initial data of Pekar product form with coherent phonon field and with the electron minimizing the corresponding energy, we provide a norm approximation of the evolution, valid up to times of order $α^2$. The approximation is given in terms of a Pekar product state, evolved through the Landau-Pekar equations, corrected by a Bogoliubov dynamics taking quantum fluctuations into account. This allows us to show that the Landau-Pekar equations approximately describe the evolution of the electron- and one-phonon reduced density matrices under the Fröhlich dynamics up to times of order $α^2$.
We consider N bosons in a box with volume one, interacting through a two-body potential with scattering length of the order $N^{-1+κ}$, for $κ>0$. Assuming that $κ\in (0;1/43)$, we show that low-energy states of the system exhibit complete Bose-Einstein condensation by providing explicit bounds on the expectation and on higher moments of the number of excitations.
We consider systems of N bosons trapped on the two-dimensional unit torus, in the Gross-Pitaevskii regime, where the scattering length of the repulsive interaction is exponentially small in the number of particles. We show that low-energy states exhibit complete Bose-Einstein condensation, with almost optimal bounds on the number of orthogonal excitations.
We review some recent results on interacting Bose gases in thermal equilibrium. In particular, we study the convergence of the grand-canonical equilibrium states of such gases to their mean-field limits, which are given by the Gibbs measures of classical field theories with quartic Hartree-type self-interaction, and to the Gibbs states of classical gases of point particles. We discuss various open problems and conjectures concerning, e.g., Bose-Einstein condensation, polymers and $\vert \boldsymbolϕ \vert^{4}$-theory.
We consider a Bose gas trapped in the unit torus in the Gross-Pitaevskii regime. In the ground state, we prove that fluctuations of bounded one-particle observables satisfy a central limit theorem.
We consider systems of bosons trapped in a box, in the Gross-Pitaevskii regime. We show that low-energy states exhibit complete Bose-Einstein condensation with an optimal bound on the number of orthogonal excitations. This extends recent results obtained in \cite{BBCS1}, removing the assumption of small interaction potential.
While Hartree-Fock theory is well established as a fundamental approximation for interacting fermions, it has been unclear how to describe corrections to it due to many-body correlations. In this paper we start from the Hartree-Fock state given by plane waves and introduce collective particle-hole pair excitations. These pairs can be approximately described by a bosonic quadratic Hamiltonian. We use Bogoliubov theory to construct a trial state yielding a rigorous Gell-Mann-Brueckner-type upper bound to the ground state energy. Our result justifies the random phase approximation in the mean-field scaling regime, for repulsive, regular interaction potentials.
We prove that Gibbs measures of nonlinear Schrödinger equations arise as high-temperature limits of thermal states in many-body quantum mechanics. Our results hold for defocusing interactions in dimensions $d =1,2,3$. The many-body quantum thermal states that we consider are the grand canonical ensemble for $d = 1$ and an appropriate modification of the grand canonical ensemble for $d =2,3$. In dimensions $d =2,3$, the Gibbs measures are supported on singular distributions, and a renormalization of the chemical potential is necessary. On the many-body quantum side, the need for renormalization is manifested by a rapid growth of the number of particles. We relate the original many-body quantum problem to a renormalized version obtained by solving a counterterm problem. Our proof is based on ideas from field theory, using a perturbative expansion in the interaction, organized by using a diagrammatic representation, and on Borel resummation of the resulting series.