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
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