Magnetic barriers of compact support and eigenvalues in spectral gaps

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.

Keywords

• Schrodinger operator
• magnetic field
• eigenvalues
• spectral gaps
• strong coupling.