Local Uniqueness and Refined Spike Profiles of Ground States for Two-Dimensional Attractive Bose-Einstein Condensates

TL;DR

This paper investigates the precise behavior and uniqueness of ground states in two-dimensional attractive Bose-Einstein condensates as the interaction strength approaches a critical value, revealing refined spike profiles and local uniqueness under specific conditions.

Contribution

It establishes the local uniqueness and detailed spike profiles of ground states near the critical interaction strength for certain trapping potentials.

Findings

Ground states are locally unique near the critical interaction strength.

Refined spike profiles of ground states are characterized as the interaction approaches the critical value.

Conditions on the trapping potential ensure non-degenerate critical points for analysis.

Abstract

We consider ground states of two-dimensional Bose-Einstein condensates in a trap with attractive interactions, which can be described equivalently by positive minimizers of the $L^2-$critical constraint Gross-Pitaevskii energy functional. It is known that ground states exist if and only if $a< a^*:= \|w\|_2^2$, where $a$ denotes the interaction strength and $w$ is the unique positive solution of $\Delta w-w+w^3=0$ in $R^2$. In this paper, we prove the local uniqueness and refined spike profiles of ground states as $a\nearrow a^*$, provided that the trapping potential $h(x)$ is homogeneous and $H(y)=\int_{R^2} h(x+y)w^2(x)dx$ admits a unique and non-degenerate critical point.