The excitation spectrum of Bose gases interacting through singular potentials

Abstract

We consider systems of N bosons in a box with volume one, interacting through a repulsive two-body potential of the form κN 3β−1 V (N β x). For all 0 < β < 1, and for sufficiently small coupling constant κ > 0, we establish the validity of Bogoliubov theory, identifying the ground state energy and the low-lying excitation spectrum up to errors that vanish in the limit of large N .

Central limit theorem for Bose gases interacting through singular potentials

“…It is proven that the ground state satisfies a central limit theorem. The arguments of the proof can be adapted to the intermediate regime β < 1 using the norm approximation for the ground state obtained in [8]. Now, we consider the time evolution of the initial data ψ N,0 = U * ϕ 0 1 ≤N T * N,0 Ω with respect to the Schrödinger equation (1.2) and show the validity of a (multi-variate) central limit theorem.…”

“…following from Theorem 1.2 for time t = 0. Note that initial data of the form ψ N,0 = U * ϕ 0 1 ≤N T * N,0 Ω describe approximate ground states of trapped systems [8]. In experiments such initial data are prepared by trapping particles through external fields and by cooling them down to extremely low temperatures so that the system essentially relaxes to its ground state.…”

Bogoliubov theory for dilute Bose gases: The Gross-Pitaevskii regime

“…On the other side for β ≥ 1/2 Bogoliubov approximation fails. Nevertheless in [9] the predictions of Bogoliubov theory where rigorously justified for any 0 < β < 1 (the proof in [9] holds for κ sufficiently small, but can be extended to any κ using the strategy recently developed for the Gross-Pitaevskii regime in [10]). The key idea to achieve this result was to understand the emergence of the scattering length as a consequence of correlations among the particles.…”

A Simple 2nd Order Lower Bound to the Energy of Dilute Bose Gases

“…The recent work [1,2] is also very relevant for (1.8), though they address the confined case in the Gross-Pitaevskii limit and not the thermodynamic limit. Actually, the result obtained in [1] is after scaling very analogous to our analysis of the box Hamiltonian (see Theorem 6.1 below). We have the additional difficulties that for our localized problem, we no longer have translation invariance nor a fixed number of particles.…”

“…We have the additional difficulties that for our localized problem, we no longer have translation invariance nor a fixed number of particles. Nevertheless, we believe that our method, at least for the ground state energy, is substantially shorter and simpler than the one of [1], which also covers the excitation spectrum.…”