Asymptotic expansion of low-energy excitations for weakly interacting bosons

Robert Seiringer,Sören Petrat,Lea Boßmann

doi:10.1017/fms.2021.22

Abstract

AbstractWe consider a system ofNbosons in the mean-field scaling regime for a class of interactions including the repulsive Coulomb potential. We derive an asymptotic expansion of the low-energy eigenstates and the corresponding energies, which provides corrections to Bogoliubov theory to any order in$1/N$.

Highlights

We consider a system of Hamiltonian interacting bosons in R,≥ 1, which are described by the -body = −Δ + ext + − =1 1≤ < ≤ (1.1)with coupling parameter1 := − 1, corresponding to a mean-field regime of weak and long-range interactions
We derive an asymptotic expansion of the low-energy eigenstates and the corresponding energies, which provides corrections to Bogoliubov theory to any order in 1/
We prove a more general statement, which can be understood as asymptotic expansion of the ground state of : for any operator ( ) on H that is relatively bounded with respect to

Summary

Introduction

1 := − 1 , corresponding to a mean-field (or Hartree) regime of weak and long-range interactions. Another approach was proposed by Pizzo in [59, 60, 61], where he considers a Bose gas on a torus in the mean-field regime He constructs an expansion for the ground state and a fixed-point equation for the ground-state energy, first for a simpler three-mode Bogoliubov Hamiltonian [59] and subsequently, building on these results, for a Bogoliubov Hamiltonian [60] and the full Hamiltonian [61]. We note that there are many recent results on the derivation of the Bogoliubov dynamics in the mean-field regime [30, 31, 35, 45], as well as in more singular scaling limits [28, 8, 49, 29, 33, 50, 20, 14, 58].

Assumptions

Lea Boßmann et al can be decomposed into condensate and excitations as

Main results

Strategy of proof

Bogoliubov theory

Bogoliubov transformations

Auxiliary estimates

Resolvent estimates