Existence, structure, and robustness of ground states of a NLSE in 3D with a point defect

TL;DR

This paper investigates the existence, structure, and robustness of ground states in a 3D nonlinear Schrödinger equation with a point defect, revealing their universal existence and specific properties regardless of interaction type.

Contribution

It proves the existence and describes the properties of ground states for a 3D NLSE with a point defect, regardless of the defect's attractive or repulsive nature.

Findings

Ground states exist for all mass values.

Ground states are positive, radially symmetric, and decreasing.

Ground states exhibit Coulombian singularity at the defect location.

Abstract

We study the ground states for the Schr\"odinger equation with a focusing nonlinearity and a point interaction in dimension three. We establish that ground states exist for every value of the mass; moreover they are positive, radially symmetric, decreasing along the radial direction, and show a Coulombian singularity at the location of the point interaction. Remarkably, the existence of the ground states is independent of the attractive or repulsive character of the point interaction.