Geodesic
Magnetic barriers of compact support and eigenvalues in spectral gaps
Electronic journal of differential equations, Tome 2003 (2003)
We consider Schrödinger 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.
Classification : 35J10, 81Q10, 35P20
Keywords: Schrödinger operator, magnetic field, eigenvalues, spectral gaps, strong coupling 
@article{EJDE_2003__2003__a145,
     author = {Hempel,  Reiner and Besch,  Alexander},
     title = {Magnetic barriers of compact support and eigenvalues in spectral gaps},
     journal = {Electronic journal of differential equations},
     year = {2003},
     volume = {2003},
     zbl = {1037.35063},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2003__2003__a145/}
}
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%A Hempel,  Reiner
%A Besch,  Alexander
%T Magnetic barriers of compact support and eigenvalues in spectral gaps
%J Electronic journal of differential equations
%D 2003
%V 2003
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Hempel,  Reiner; Besch,  Alexander. Magnetic barriers of compact support and eigenvalues in spectral gaps. Electronic journal of differential equations, Tome 2003 (2003). http://geodesic.mathdoc.fr/item/EJDE_2003__2003__a145/