The excitation spectrum of a Bose gas with an impurity in the Gross-Pitaevskii regime

TL;DR

This paper analyzes the excitation spectrum of a dilute Bose gas with an impurity, deriving energy expansions and connecting eigenvalues to a renormalized Hamiltonian in the large particle limit.

Contribution

It provides a rigorous derivation of the ground state energy expansion and characterizes the excitation spectrum via a renormalized Bogoliubov--Fr"ohlich Hamiltonian.

Findings

Ground state energy expanded up to order one in particle number

Excited eigenvalues correspond to the eigenvalues of a renormalized Hamiltonian

Results valid in the large particle number limit

Abstract

We study a dilute system of $N$ interacting bosons coupled to an impurity particle via a pair potential in the Gross--Pitaevskii regime. We derive an expansion of the ground state energy up to order one in the boson number, and show that the difference of excited eigenvalues to the ground state is given by the eigenvalues of the renormalized Bogoliubov--Fr\"ohlich Hamiltonian in the limit $N\to \infty$.