Sharp Spectral Asymptotics for two-dimensional Schrödinger operator with a strong degenerating magnetic field

Details Sharp Spectral Asymptotics for two-dimensional Schrödinger operator with a strong degenerating magnetic field Sharp Spectral Asymptotics for two-dimensional Schrödinger operator with a strong degenerating magnetic field

2006-03-04

arxiv.org/abs/math/0603114v1

Mathematics

Analysis of PDEs

79 pages

Scientific paper

I consider two-dimensional Schr\"odinger operator with degenerating magnetic field and in the generic situation I derive spectral asymptotics as $h\to +0$ and $\mu\to +\infty$ where $h$ and $\mu$ are Planck and coupling parameters respectively. The remainder estimate is $O(\mu^{-{\frac 1 2}}h^{-1})$ which is between $O(\mu^{-1}h^{-1})$ valid as magnetic field non-degenerates and $O(h^{-1})$ valid as magnetic field is identically 0. As $\mu$ is close to its maximal reasonable value $O(h^{-2})$ the principal part contains correction terms associated with short periodic trajectories of the corresponding classical dynamics.

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