On the Mass Concentration for Bose-Einstein Condensates with Attractive Interactions

TL;DR

This paper analyzes how the mass of two-dimensional Bose-Einstein condensates with attractive interactions concentrates at the global minimum of the trapping potential as the interaction strength approaches a critical value, using the Gross-Pitaevskii functional.

Contribution

It provides a detailed mathematical analysis of the mass concentration phenomenon for minimizers of the Gross-Pitaevskii functional near the critical interaction strength in 2D.

Findings

Mass concentrates at the global minimum as interaction approaches critical value

Existence of minimizers only if interaction strength is below a critical threshold

Characterization of the asymptotic behavior of minimizers near the critical point

Abstract

We consider two-dimensional Bose-Einstein condensates with attractive interaction, described by the Gross-Pitaevskii functional. Minimizers of this functional exist only if the interaction strength $a$ satisfies $a < a^*= \|Q\|_2^2$, where $Q$ is the unique positive radial solution of $\Delta u-u+u^3=0$ in $\R^2$. We present a detailed analysis of the behavior of minimizers as $a$ approaches $a^*$, where all the mass concentrates at a global minimum of the trapping potential.