Heat kernel of integrable billiards in a magnetic field

TL;DR

This paper develops analytical methods to compute the magnetic response of electrons in billiard domains, comparing two approaches and discussing their agreement and differences with previous results.

Contribution

It introduces two analytical methods for calculating the heat kernel and magnetic response in integrable billiards, extending previous work and analyzing their applicability.

Findings

Both methods agree with each other but differ from Robnik's results.

The Balian-Bloch multiple scattering expansion is analyzed.

Extensions to other geometries are discussed.

Abstract

We present analytical methods to calculate the magnetic response of non-interacting electrons constrained to a domain with boundaries and submitted to a uniform magnetic field. Two different methods of calculation are considered - one involving the large energy asymptotic expansion of the resolvent (Stewartson-Waechter method) is applicable to the case of separable systems, and another based on the small time asymptotic behaviour of the heat kernel (Balian-Bloch method). Both methods are in agreement with each other but differ from the result obtained previously by Robnik. Finally, the Balian-Bloch multiple scattering expansion is studied and the extension of our results to other geometries is discussed.