Quasi-classical versus non-classical spectral asymptotics for magnetic Schroedinger operators with decreasing electric potentials

TL;DR

This paper studies the spectral asymptotics of magnetic Schrödinger operators with rapidly decaying electric potentials, revealing non-classical behavior when the potential decays faster than polynomially.

Contribution

It demonstrates that for exponentially decaying potentials, the spectral asymptotics deviate from classical quasi-classical formulas, highlighting new non-classical spectral phenomena.

Findings

Spectral asymptotics differ from classical predictions for Gaussian decay.

Non-classical behavior occurs when the potential decays faster than polynomial.

Results apply to Schrödinger operators with constant magnetic fields in 2D and 3D.

Abstract

We consider the Schroedinger operator H on L^2(R^2) or L^2(R^3) with constant magnetic field and electric potential V which typically decays at infinity exponentially fast or has a compact support. We investigate the asymptotic behaviour of the discrete spectrum of H near the boundary points of its essential spectrum. If the decay of V is Gaussian or faster, this behaviour is non-classical in the sense that it is not described by the quasi-classical formulas known for the case where V admits a power-like decay.