Fundamental gaps of the Gross-Pitaevskii equation with repulsive interaction

TL;DR

This paper investigates the fundamental energy and chemical potential gaps in the Gross-Pitaevskii equation with repulsive interactions, providing asymptotic and numerical insights, and proposing a conjecture on these gaps under various boundary conditions.

Contribution

It offers new asymptotic and numerical analysis of the fundamental gaps in the GPE and formulates a conjecture on their behavior under different trapping potentials and boundary conditions.

Findings

Formulated a gap conjecture for bounded domains with Dirichlet boundary conditions.

Extended results to Neumann and periodic boundary conditions.

Provided asymptotic and numerical evidence for gap behaviors.

Abstract

We study asymptotically and numerically the fundamental gaps (i.e. the difference between the first excited state and the ground state) in energy and chemical potential of the Gross-Pitaevskii equation (GPE) -- nonlinear Schrodinger equation with cubic nonlinearity -- with repulsive interaction under different trapping potentials including box potential and harmonic potential. Based on our asymptotic and numerical results, we formulate a gap conjecture on the fundamental gaps in energy and chemical potential of the GPE on bounded domains with the homogeneous Dirichlet boundary condition, and in the whole space with a convex trapping potential growing at least quadratically in the far field. We then extend these results to the GPE on bounded domains with either the homogeneous Neumann boundary condition or periodic boundary condition.