Energy estimates and symmetry breaking in attractive Bose-Einstein condensates with ring-shaped potentials

TL;DR

This paper investigates the energy minimizers of a two-dimensional Bose-Einstein condensate with attractive interactions in a ring-shaped potential, revealing symmetry breaking at critical interaction strength and conditions for uniqueness and symmetry.

Contribution

It establishes precise energy estimates and demonstrates symmetry breaking and uniqueness of minimizers in the Gross-Pitaevskii model with ring-shaped potentials.

Findings

Symmetry breaking occurs as interaction strength approaches a critical value.

Minimizers concentrate on the circular bottom of the potential well.

For small interaction strength, minimizers are unique and radially symmetric.

Abstract

This paper is concerned with the properties of $L^2$-normalized minimizers of the Gross-Pitaevskii (GP) functional for a two-dimensional Bose-Einstein condensate with attractive interaction and ring-shaped potential. By establishing some delicate estimates on the least energy of the GP functional, we prove that symmetry breaking occurs for the minimizers of the GP functional as the interaction strength $a>0$ approaches a critical value $a^*$, each minimizer of the GP functional concentrates to a point on the circular bottom of the potential well and then is non-radially symmetric as $a\nearrow a^*$. However, when $a>0$ is suitably small we prove that the minimizers of the GP functional are unique, and this unique minimizer is radially symmetric.