Quantum wires with magnetic fluxes

TL;DR

This paper analyzes magnetic Laplacians on graphs, describing how magnetic fluxes influence the operators and establishing conditions for their unique determination by fluxes, with implications for quantum graph theory.

Contribution

It provides a complete characterization of magnetic Laplacians on graphs and conditions for their unique specification by fluxes, extending understanding of magnetic effects in quantum graphs.

Findings

The set of magnetic Laplacians is generically isomorphic to a torus.

Conditions are identified for unique operator determination by fluxes.

A comprehensive description of magnetic perturbations on graph Laplacians.

Abstract

In the present article magnetic Laplacians on a graph are analyzed. We provide a complete description of the set of all operators which can be obtained from a given self-adjoint Laplacian by perturbing it by magnetic fields. In particular, it is shown that generically this set is isomorphic to a torus. We also describe the conditions under which the operator is unambiguously (up to unitary equivalence) defined by prescribing the magnetic fluxes through all loops of the graph.