- 07-20 Sylvain Golenia, Sergiu Moroianu
- Spectral analysis of magnetic Laplacians on conformally
Jan 26, 07
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Abstract. We consider an open manifold which is the interior of a compact manifold
with boundary. Assuming gauge invariance, we classify magnetic fields
with compact support into being trapping or non-trapping.
We study spectral properties of the associated magnetic Laplacian for a
class of Riemannian metrics which includes complete hyperbolic metrics of
When $B$ is non-trapping, the magnetic Laplacian has nonempty
essential spectrum. Using Mourre theory, we show the absence of
singular continuous spectrum and the local finiteness of the point
spectrum. When $B$ is trapping, the spectrum is discrete and obeys
the Weyl law. The existence of trapping magnetic fields with compact
support depends on cohomological conditions, indicating a new and
very strong long-range effect.
In the non-gauge invariant case, we exhibit a strong Aharonov-Bohm effect.
On hyperbolic surfaces with at least two cusps, we show
that the magnetic Laplacian associated to every magnetic field with
compact support has purely discrete spectrum for some choices of the vector
potential, while other choices lead to a situation of limit absorption
We also study perturbations of the metric.
We show that in the Mourre theory it is not necessary to require
a decay of the derivatives of the perturbation. This very singular
perturbation is then brought closer to the perturbation
of a potential.