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.