Ground states for the NLS on non-compact graphs with an attractive potential

TL;DR

This paper investigates the existence of ground states for the subcritical nonlinear Schrödinger equation on non-compact quantum graphs with an attractive potential, revealing conditions for their existence based on mass and graph features.

Contribution

It provides new insights into the existence and nonexistence of ground states on quantum graphs with attractive potentials, considering the effects of graph geometry and potential strength.

Findings

Ground states exist for small and large mass values.

Intermediate mass ranges may lack ground states depending on graph and potential.

The results are inspired by quantum waveguide studies involving curvature-induced potentials.

Abstract

We consider the subcritical nonlinear Schr\"odinger equation on non-compact quantum graphs with an attractive potential supported in the compact core, and investigate the existence and the nonexistence of Ground States, defined as minimizers of the energy at fixed $L^2$-norm, or mass. We finally reach the following picture: for small and large mass there are Ground States. Moreover, according to the metric features of the compact core of the graph and to the strength of the potential, there may be an interval of intermediate masses for which there are no Ground States. The study was inspired by the research on quantum waveguides, in which the curvature of a thin tube induces an effective attractive potential.