Fundamental gaps of the Gross–Pitaevskii equation with repulsive interaction

Article type: Research Article

Authors: Bao, Weizhu^a | Ruan, Xinran^{a; b; *}

Affiliations: [a] Department of Mathematics, National University of Singapore, Singapore 119076. E-mail: matbaowz@nus.edu.sg | [b] Laboratoire J.-L. Lions, Université Pierre et Marie Curie, Paris cedex 05, France. E-mails: ruan@ljll.math.upmc.fr, a0103426@u.nus.edu

Correspondence: [*] Corresponding author. E-mail: ruan@ljll.math.upmc.fr.

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 Schrödinger 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.

Keywords: Gross–Pitaevskii equation, fundamental gap, ground state, first excited state, energy asymptotics, repulsive interaction

DOI: 10.3233/ASY-181477

Journal: Asymptotic Analysis, vol. 110, no. 1-2, pp. 53-82, 2018