Large time behavior of the heat kernel of two-dimensional magnetic Schroedinger operators

TL;DR

This paper investigates the long-term behavior of the heat kernel for two-dimensional magnetic Schrödinger operators, revealing the influence of magnetic flux and providing explicit formulas for specific magnetic fields.

Contribution

It establishes the large time asymptotics of the heat kernel based on magnetic flux and derives exact formulas for Aharonov-Bohm magnetic fields.

Findings

Large time behavior depends on total magnetic flux for radial fields

Provides on-diagonal heat kernel estimates

Derives explicit heat kernel formulas for Aharonov-Bohm field

Abstract

We study the heat semigroup generated by two-dimensional Schroedinger operators with compactly supported magnetic field. We show that if the field is radial, then the large time behavior of the associated heat kernel is determined by its total flux. We also establish some on-diagonal heat kernel estimates and discuss their applications for solutions to the heat equation. An exact formula for the heat kernel, and for its large time asymptotic, is derived in the case of the Aharonov-Bohm magnetic field.