Ground states of the defocusing nonlinear Schr\"{o}dinger equation with a point interaction in dimensions 2 and 3

TL;DR

This paper investigates the existence and properties of ground states for the defocusing nonlinear Schrödinger equation with a point interaction in two and three dimensions, focusing on small mass regimes and specific parameter conditions.

Contribution

It establishes the existence of ground states at small masses and explores their qualitative properties and relation to critical points of the action functional.

Findings

Ground states exist for small masses in specified parameter regimes.

Qualitative properties of these ground states are characterized.

Connections between ground states and critical points of the action functional are identified.

Abstract

This paper is concerned with ground states of the defocusing nonlinear Schr\"odinger equation with a point interaction, \[ \mathrm{i} \partial_t \psi = -\Delta_\alpha \psi + \psi |\psi|^{p - 2} \quad \text{in} \quad \mathbb{R} \times \mathbb{R}^N, \] where $- \Delta_\alpha$ denotes the Laplacian of point interaction centered at the origin with inverse s-wave scattering length $- 2 (N - 1) \pi \alpha$ and we suppose that either (i) $N = 2$, $\alpha \in \mathbb{R}$ and $p > 2$ or (ii) $N = 3$, $\alpha < 0$ and $2 < p < 3$. At sufficiently small masses, (i) we prove that this equation admits ground states, (ii) we obtain some qualitative properties of ground states and (iii) we obtain some results relating ground states with critical points of the associated action functional.