Nonlinear Schr\"odinger Equation with magnetic potential on metric graphs

TL;DR

This paper studies the nonlinear magnetic Schr"odinger equation on metric graphs, showing how magnetic effects influence ground state existence and structure, including phase transitions on specific graph topologies.

Contribution

It introduces a reduction of the magnetic problem to a non-magnetic one with effective potentials, extending classical existence criteria to magnetic settings.

Findings

Ground states exist for small repulsion and specific mass ranges.

Strong magnetic flux can prevent ground state formation.

A phase transition in ground state structure is characterized on the tadpole graph.

Abstract

In this manuscript, we shall investigate the Nonlinear Magnetic Schr\"odinger Equation on noncompact metric graphs, focusing on the existence of ground states. We prove that the magnetic Hamiltonian is variationally equivalent to a non-magnetic operator with additional repulsive potentials supported on the graph's cycles. This effective potential is strictly determined by the Aharonov-Bohm flux through the topological loops. Leveraging this reduction, we extend classical existence criteria to the magnetic setting. As a key application, we characterize the ground state structure on the tadpole graph, revealing a mass-dependent phase transition. The ground states exist for sufficiently small repulsion in an intermediate regime of masses while sufficiently strong flux prevents the formation of ground states.