Time dependent Schr\"odinger equation for harmonic oscillator in the Aharonov-Bohm magnetic field

TL;DR

This paper develops an approximation for the solution kernel of the 2D harmonic oscillator Schr"odinger equation in an Aharonov-Bohm magnetic field, utilizing Fourier Integral Operators and deriving a Mehler formula variant.

Contribution

It introduces a novel approximation method for the Schr"odinger kernel in a magnetic field setting, extending Fourier Integral Operator techniques.

Findings

Derived a main term approximation of the kernel matching a modified Mehler formula.

Extended Fourier Integral Operator methods to a magnetic field context.

Provided analytical tools for quantum systems with magnetic flux.

Abstract

We construct an approximation of the kernel of the solution of the time dependent Schr\"odinger equation whose Hamiltonian is a 2D harmonic oscillator in Aharonov-Bohm magnetic field. The main tools used here were established in the paper of A. Laptev and I.M. Sigal, where the authors considered a class of Fourier Integral Operators with global complex phases approximating the fundamental solutions (propagators) for time-dependent Schr\"odinger equations. For the example considered in this paper we are able to find the main term in the approximation of the kernel that equals a version of the Mehler formula.