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Nonlinear Schrödinger Equation with magnetic potential on metric graphs

Riccardo Adami, Nicolò Cangiotti, Ivan Gallo, David Spitzkopf

TL;DR

This work analyzes the magnetic nonlinear Schrödinger equation on noncompact metric graphs, establishing that the magnetic problem is variationally equivalent to a non-magnetic NLSE with an effective repulsive potential supported on graph cycles, whose strength is determined by the Aharonov-Bohm flux through those cycles. A explicit formula $\Phi_{\gamma}(A) = \frac{4\pi^2}{|\gamma|^2} \mathrm{dist}\left( \frac{\int_{\gamma} A dx}{2\pi}, \mathbb{Z} \right)^2$ captures the flux contribution, enabling a cycle-wise reduction to the functional $I_A(v,\mathcal{G})$ and the extension of classical existence criteria via concentration-compactness. The theory is applied to the tadpole graph, where existence holds in an intermediate mass range under small flux, and nonexistence occurs for sufficiently strong noninteger flux; a hyperbolic-secant competitor yields explicit, verifiable conditions for the existence region. Overall, the paper provides a rigorous variational framework for magnetic effects on quantum graphs and delivers concrete existence/nonexistence criteria for ground states, including explicit constructions in the pivotal tadpole geometry.

Abstract

In this manuscript, we shall investigate the Nonlinear Magnetic Schrödinger 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.

Nonlinear Schrödinger Equation with magnetic potential on metric graphs

TL;DR

This work analyzes the magnetic nonlinear Schrödinger equation on noncompact metric graphs, establishing that the magnetic problem is variationally equivalent to a non-magnetic NLSE with an effective repulsive potential supported on graph cycles, whose strength is determined by the Aharonov-Bohm flux through those cycles. A explicit formula captures the flux contribution, enabling a cycle-wise reduction to the functional and the extension of classical existence criteria via concentration-compactness. The theory is applied to the tadpole graph, where existence holds in an intermediate mass range under small flux, and nonexistence occurs for sufficiently strong noninteger flux; a hyperbolic-secant competitor yields explicit, verifiable conditions for the existence region. Overall, the paper provides a rigorous variational framework for magnetic effects on quantum graphs and delivers concrete existence/nonexistence criteria for ground states, including explicit constructions in the pivotal tadpole geometry.

Abstract

In this manuscript, we shall investigate the Nonlinear Magnetic Schrödinger 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.
Paper Structure (8 sections, 11 theorems, 82 equations, 3 figures)

This paper contains 8 sections, 11 theorems, 82 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Magnetic NLSE framework
  3. Preliminaries and Functional Framework
  4. Magnetic Sobolev space
  5. Effective Repulsion and Variational Reduction
  6. Existence of ground states
  7. The tadpole graph
  8. The hyperbolic secant competitor

Key Result

Proposition 3.1

Let $\mathcal{G}$ be a noncompact metric graph and let $p>2$. Then there exists a constant $K_{\mathcal{G},p}>0$ such that for all $u \in H^1(\mathcal{G})$:

Figures (3)

  • Figure 1: The tadpole graph.
  • Figure 2: Competitor function $u_\mu(x)$ with $L=1$ and $\mu=1$
  • Figure 3: Parameter region in the $(\mu, \phi_\gamma)$-plane where the condition of the inequality \ref{['magneticexistence']} holds with $L=1$.

Theorems & Definitions (26)

  • Proposition 3.1: Gagliardo-Nirenberg
  • Corollary 3.2
  • Definition 3.3
  • Definition 3.4
  • Remark 3.5
  • Theorem 3.6: Diamagnetic inequality on metric graphs
  • proof
  • Proposition 4.1
  • proof
  • Remark 4.2
  • ...and 16 more