The
Neumann realization for the Schrödinger operator with magnetic
field is considered in a bounded two-dimensional domain with corners.
This operator is associated with a small semi-classical parameter
h or, equivalently, with a large magnetic field.
We investigate the behavior of its eigenpairs as h
tends to zero, like in a semi-classical limit. We prove, in the
situation where the domain is a polygon and the magnetic field is
constant, that the lowest eigenvalues are exponentially close to those
of model problems associated with the corners. We approximate the
corresponding eigenvectors by linear combinations of functions
concentrated in corners at the scale \sqrt(h). An oscillatory term at the scale h multiplies these latter functions.
If the domain has curved sides and the magnetic field is smoothly
varying, we exhibit a full asymptotics for eigenpairs in powers
of \sqrt(h).
14 October 2005. Preprint IRMAR 05-27.
Annales Henri Poincaré Vol. 7 (2006), pp 899-931.
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