Extensões auto-adjuntas do operador de Schrödinger magnético em domínios com pouca regularidade

Abstract

We find one parametrization of all self-adjoint extensions of the magnetic Schrödinger operator, in a quasi-convex domain with compact boundary, and magnetic potentials with low regularity. In this parametrization we use boundary-triples, which also gives a new characterization of all self-adjoint extensions of the Laplacian in quasi-convex domains. Then we discuss gauge transformations for such self-adjoint extensions and generalize, for all self-adjoint extensions, a characterization, due to Helffer, of the gauge equivalence of the Dirichlet magnetic operator with 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