Positive-density ground states of the Gross-Pitaevskii equation

TL;DR

This paper investigates the existence and phase transition of positive-density ground states in the Gross-Pitaevskii equation, revealing conditions under which constant solutions are no longer energy minimizers at high density.

Contribution

It establishes the occurrence of a phase transition in the ground states of the Gross-Pitaevskii equation based on the Fourier transform of the interaction potential.

Findings

Existence of a phase transition at high density when the Fourier transform of the potential is negative.

Constant solutions cease to be ground states beyond a critical density.

The analysis combines elliptic PDE and statistical mechanics techniques.

Abstract

We consider the nonlinear Gross-Pitaevskii equation at positive density, that is, for a bounded solution not tending to 0 at infinity. We focus on infinite ground states, which are by definition minimizers of the energy under local perturbations. When the Fourier transform of the interaction potential takes negative values we prove the existence of a phase transition at high density, where the constant solution ceases to be a ground state. The analysis requires mixing techniques from elliptic PDE theory and statistical mechanics, in order to deal with a large class of interaction potentials.