Normalized solutions of nonlinear magnetic Schr\"odinger equations on metric graphs

TL;DR

This paper develops a magnetic Sobolev space framework on metric graphs, proves self-adjointness of the magnetic Schr"odinger operator, and investigates the existence and multiplicity of normalized solutions for nonlinear magnetic Schr"odinger equations across various graph types and nonlinear regimes.

Contribution

It introduces a magnetic Sobolev space on metric graphs and analyzes normalized solutions of nonlinear magnetic Schr"odinger equations, covering multiple graph and nonlinearity types.

Findings

Established magnetic Sobolev space $H^1_A(\

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Abstract

In this paper we first establish the theory of a magnetic Sobolev space $H^1_A(\mathcal{G},\mathbb{C})$ on metric graphs $\mathcal{G}$ and we prove the self-adjointness of its corresponding magnetic Schr\"odinger operator. Then, in this setting, we investigate the existence and multiplicity of normalized solutions to nonlinear magnetic Schr\"odinger equations on compact metric graphs and on noncompact metric graphs with localized nonlinearities or nonlinearities acting on whole metric graphs, covering the mass-subcritical, mass-critical, and mass-supercritical cases.