Electron. J. Diff. Eqns., Vol. 2003(2003), No. 48, pp. 1-25.

Magnetic barriers of compact support and eigenvalues in spectral gaps

Rainer Hempel & Alexander Besch

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.

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Rainer Hempel
Institut fur Analysis
Technische Universitat Braunschweig
Pockelsstrasse 14, 38106 Braunschweig
Germany
email: r.hempel@tu-bs.de
Alexander Besch
Institut fur Analysis
Technische Universitat Braunschweig
Pockelsstrasse 14, 38106 Braunschweig
Germany
Current address: Volkswagen AG, Wolfsburg, Germany
email: alexander.besch@volkswagen.de

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