Eigenvalues of a nonlinear ground state in the Thomas-Fermi approximation

TL;DR

This paper investigates the eigenvalues of a nonlinear ground state in the Thomas-Fermi approximation of the Gross-Pitaevskii equation, providing convergence analysis, resolvent estimates, and asymptotic eigenfunction approximations.

Contribution

It offers a rigorous analysis of eigenvalue convergence, resolvent norm convergence, and asymptotic eigenfunction approximations in the hydrodynamics limit.

Findings

Proved convergence of eigenvalues in the hydrodynamics limit.

Estimated the convergence rate of the resolvent operator.

Provided asymptotic and numerical approximations of eigenfunctions and eigenvalues.

Abstract

We study a nonlinear ground state of the Gross-Pitaevskii equation with a parabolic potential in the hydrodynamics limit often referred to as the Thomas--Fermi approximation. Existence of the energy minimizer has been known in literature for some time but it was only recently when the Thomas-Fermi approximation was rigorously justified. The spectrum of linearization of the Gross-Pitaevskii equation at the ground state consists of an unbounded sequence of positive eigenvalues. We analyze convergence of eigenvalues in the hydrodynamics limit. Convergence in norm of the resolvent operator is proved and the convergence rate is estimated. We also study asymptotic and numerical approximations of eigenfunctions and eigenvalues using Airy functions.