All self-adjoint extensions of the magnetic Laplacian in nonsmooth domains and gauge transformations

TL;DR

This paper characterizes all self-adjoint extensions of the magnetic Laplacian in nonsmooth domains using boundary triples, explores gauge transformations, and relates these to the Aharonov-Bohm effect, extending previous results to irregular domains.

Contribution

It provides a comprehensive parametrization of all self-adjoint extensions of the magnetic Schrödinger operator in nonsmooth domains and analyzes gauge equivalences, including irregular solenoids.

Findings

All self-adjoint extensions are parametrized via boundary triples.

Gauge transformations are characterized for these extensions.

Unitary equivalence to zero magnetic potential extends to all realizations in quasi-convex domains.

Abstract

We use boundary triples to find a parametrization of all self-adjoint extensions of the magnetic Schr\"odinger operator, in a quasi-convex domain~$\Omega$ with compact boundary, and magnetic potentials with components in $\textrm{W}^{1}_{\infty}(\overline{\Omega})$. This gives also a new characterization of all self-adjoint extensions of the Laplacian in nonregular domains. Then we discuss gauge transformations for such self-adjoint extensions and generalize a characterization of the gauge equivalence of the Dirichlet magnetic operator for the Dirichlet Laplacian; the relation to the Aharonov-Bohm effect, including irregular solenoids, is also discussed. In particular, in case of (bounded) quasi-convex domains it is shown that if some extension is unitarily equivalent (through the multiplication by a smooth unit function) to a realization with zero magnetic potential, then the same occurs for all self-adjoint realizations.