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

2006/03/04 by Ivrii, Victor
#35P20 #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.math/0603114

Abstract

I consider two-dimensional Schrödinger operator with degenerating magnetic field and in the generic situation I derive spectral asymptotics as h→ +0 and μ→ +∞ where h and μ are Planck and coupling parameters respectively. The remainder estimate is O(μ^-\frac 1 2h^-1) which is between O(μ^-1h^-1) valid as magnetic field non-degenerates and O(h^-1) valid as magnetic field is identically 0. As μ 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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