The heat kernel for two Aharonov-Bohm solenoids in a uniform magnetic field

TL;DR

This paper derives a heat kernel formula for a quantum particle influenced by two Aharonov-Bohm solenoids and a magnetic field, providing asymptotic estimates for eigenvalues and eigenfunctions in a complex magnetic environment.

Contribution

It introduces a novel heat kernel formula for Schrödinger operators with two Aharonov-Bohm solenoids in a magnetic field, extending existing techniques to multiply connected spaces.

Findings

Derived an explicit heat kernel formula for the system.

Provided asymptotic expressions for the lowest eigenvalue.

Analyzed eigenfunctions near the Landau level.

Abstract

A non-relativistic quantum model is considered with a point particle carrying a charge $e$ and moving on the plane pierced by two infinitesimally thin Aharonov-Bohm solenoids and subjected to a perpendicular uniform magnetic field of magnitude $B$. Relying on a technique due to Schulman and Sunada which is applicable to Schr\"odinger operators on multiply connected configuration manifolds a formula is derived for the corresponding heat kernel. As an application of the heat kernel formula, an approximate asymptotic expressions are derived for the lowest eigenvalue lying above the first Landau level and for the corresponding eigenfunction while assuming that $|eB|R^{2}/(\hbar c)$ is large where $R$ is the distance between the two solenoids.