Magnetic barriers of compact support and eigenvalues in spectral gaps

• Main Authors: Rainer Hempel, Alexander Besch
• Format: Article
• Language: English
• Published: Texas State University 2003-04-01
• Series: Electronic Journal of Differential Equations
• Subjects: Schrodinger operator; magnetic field; eigenvalues; spectral gaps; strong coupling.
• Online Access: http://ejde.math.txstate.edu/Volumes/2003/48/abstr.html

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.