Ground States for the Defocusing Nonlinear Schr\"odinger Equation on Non-Compact Metric Graphs

TL;DR

This paper studies the existence and stability of ground states for the defocusing nonlinear Schrödinger equation on non-compact metric graphs, establishing spectral criteria and exploring the relationship between action and energy ground states.

Contribution

It provides a sharp spectral criterion for ground state existence and clarifies the relationship between action and energy minimizers on non-compact metric graphs.

Findings

Ground states exist if and only if the spectrum's bottom is negative and the frequency is in a specific range.

Every action minimizer is an energy minimizer, but not vice versa.

Energy ground states may not exist for certain masses, especially in the mass-subcritical case.

Abstract

We investigate the existence and stability of ground states for the defocusing nonlinear Schr\"odinger equation on non-compact metric graphs. We establish a sharp criterion for the existence of action ground states in terms of the spectral properties of the underlying Hamiltonian: ground states exist if and only if the bottom of the spectrum is negative and the frequency lies within a suitable range. We further explore the relation between action and energy ground states, showing that while every action minimizer yields an energy minimizer, the converse fails in general. In particular, we prove that energy ground states may not exist for arbitrary masses. This discrepancy is illustrated through explicit examples on star graphs with $\delta$ and $\delta'$-type vertex conditions: in the mass-subcritical case, we exhibit a large interval of masses for which no energy minimizer exists, whereas in the supercritical regime, energy ground states exist for all masses.