Spectroscopy of drums and quantum billiards: perturbative and non-perturbative results

TL;DR

This paper introduces advanced numerical and analytical methods for solving the Helmholtz equation on arbitrary domains, providing exact formulas for membrane states and a perturbative scheme for small shape deformations, with various applications.

Contribution

It presents new theorems for exact membrane state solutions and a systematic perturbative approach for shape deformations, advancing the analysis of quantum billiards and drums.

Findings

Exact formulas for ground and excited membrane states

Perturbative scheme for small shape deformations

Numerical and analytical results for various applications

Abstract

We develop powerful numerical and analytical techniques for the solution of the Helmholtz equation on general domains. We prove two theorems: the first theorem provides an exact formula for the ground state of an arbirtrary membrane, while the second theorem generalizes this result to any excited state of the membrane. We also develop a systematic perturbative scheme which can be used to study the small deformations of a membrane of circular or square shapes. We discuss several applications, obtaining numerical and analytical results.