2D- and 3DMagnetic Schroedinger Operators: Short Loops, Pointwise Spectral Asymptotics and Asymptotics of Dirac Energy

TL;DR

This paper derives pointwise spectral asymptotics for 2D and 3D magnetic Schrödinger operators, emphasizing the role of loops over periodic trajectories, and explores related asymptotic expressions under non-degeneracy conditions.

Contribution

It provides new pointwise spectral asymptotics for magnetic Schrödinger operators in 2D and 3D, highlighting the significance of loops instead of periodic trajectories.

Findings

Spectral asymptotics are derived for 2D and 3D Schrödinger operators with magnetic fields.

Loops are identified as crucial elements in the asymptotic analysis.

Related asymptotic expressions are also established.

Abstract

We consider 2- and 3-dimensional Schr\"odinger or generalized Schr\"odinger-Pauli operators with the non-degenerating magnetic field in the open domain under certain non-degeneracy assumptions we derive pointwise spectral asymptotics. We also consider asymptotics of some related expressions (see below). For all asymptotics loops rather than periodic trajectories play important role.