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

Abstract

I consider two-dimensional Schrödinger 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.

79 pages

References in corpus (2)

• Sharp Spectral Asymptotics for Operators with Irregular Coefficients. III. Schroedinger operator with a strong magnetic field
• Sharp Spectral Asymptotics for Operators with Irregular Coefficients. V. Multidimensional Schroedinger operator with a strong magnetic field. Non-Full-rank case

Cited by in corpus (5)

• Sharp Spectral Asymptotics for Operators with Irregular Coefficients. III. Schroedinger operator with a strong magnetic field
• Sharp Spectral Asymptotics for Operators with Irregular Coefficients. V. Multidimensional Schroedinger operator with a strong magnetic field. Non-Full-rank case
• Sharp Spectral Asymptotics for four-dimensional Schroedinger operator with a strong magnetic field. II
• Magnetic Schroeodinger Operator: Geometry, Classical and Quantum Dynamics and Spectral Aymptotics
• Sharp Spectral Asymptotics for Dirac Energy. II. Magnetic Schroedinger operator