Semi-classical spectrum of integrable systems in a magnetic field

TL;DR

This paper derives semi-classical energy spectra and wavefunctions for an electron in a magnetic field within integrable geometries, providing analytical solutions that match numerical results and elucidate bulk-edge state connections.

Contribution

It introduces uniform asymptotic approximations for WKB energies and wavefunctions in integrable geometries under magnetic fields, bridging classical and quantum descriptions.

Findings

Analytical solutions agree with numerical results for low-energy states

Uniform approximation effectively connects bulk and edge states

Semi-classical spectra accurately describe quantum behavior in studied geometries

Abstract

The quantum dynamics of an electron in a uniform magnetic field is studied for geometries corresponding to integrable cases. We obtain the uniform asymptotic approximation of the WKB energies and wavefunctions for the semi-infinite plane and the disc. These analytical solutions are shown to be in excellent agreement with the numerical results obtained from the Schrodinger equations even for the lowest energy states. The classically exact notions of bulk and edge states are followed to their semi-classical limit, when the uniform approximation provides the connection between bulk and edge.