Electronic Journal of Differential Equations,
Vol. 2003(2003), No. 48, pp. 1-25.
Title: Magnetic barriers of compact support and eigenvalues in
spectral gaps
Authors: Rainer Hempel (Technische Univ. Braunschweig, Germany)
Alexander Besch (Volkswagen AG, Wolfsburg, Germany)
Abstract:
We consider Schr\"odinger operators
$H = -\Delta + V$ in $L_2(\mathbb{R}^2)$
with a spectral gap, perturbed by a strong magnetic field $\mathcal{B}$
of compact support.
We assume here that the support of $\mathcal{B}$ is connected and has
a connected complement; the total magnetic flux may be zero
or non-zero. For a fixed point $E$ in the gap, we show that
(for a sequence of couplings tending to $\infty$) the signed
spectral flow across $E$ for the magnetic perturbation is equal
to the flow of eigenvalues produced by a high potential barrier
on the support of the magnetic field. This allows us to use
various estimates that are available for the high barrier case.
Submitted May 22, 2001. Published April 24, 2003.
Math Subject Classifications: 35J10, 81Q10, 35P20.
Key Words: Schrodinger operator; magnetic field; eigenvalues; spectral gaps;
strong coupling.