Global well-posedness for a nonlocal Gross-Pitaevskii equation with non-zero condition at infinity

TL;DR

This paper establishes conditions under which a nonlocal Gross-Pitaevskii equation with non-zero boundary conditions at infinity is globally well-posed across various dimensions, expanding understanding of such nonlocal quantum systems.

Contribution

It provides new sufficient conditions for global well-posedness of the nonlocal Gross-Pitaevskii equation with non-zero boundary conditions, covering a broad class of potentials.

Findings

Global well-posedness established for a variety of nonlocal interactions.

Conditions applicable to even, positive definite, or positive tempered distribution potentials.

Results extend to any spatial dimension.

Abstract

We study the Gross-Pitaevskii equation involving a nonlocal interaction potential. Our aim is to give sufficient conditions that cover a variety of nonlocal interactions such that the associated Cauchy problem is globally well-posed with non-zero boundary condition at infinity, in any dimension. We focus on even potentials that are positive definite or positive tempered distributions.