Eigenvalue Problem in Two Dimension for An Irregular Boundary

TL;DR

This paper introduces an analytical perturbative approach to solve the Helmholtz equation in two dimensions with irregular boundaries, enabling calculation of energy levels and vibration frequencies for complex shapes.

Contribution

The paper presents a novel perturbative method for solving the eigenvalue problem in irregular two-dimensional domains, extending analytical solutions to complex boundary shapes.

Findings

Method accurately computes energy levels for elliptical and supercircular boundaries.

Results agree with numerical calculations, validating the approach.

Discusses shape-induced level crossing phenomena.

Abstract

An analytical perturbative method is suggested for solving the Helmholtz equation (\bigtriangledown^{2} + k^{2}){\psi} = 0 in two dimensions where {\psi} vanishes on an irregular closed curve. We can thus find the energy levels of a quantum mechanical particle confined in an infinitely deep potential well in two dimensions having an irregular boundary or the vibration frequencies of a membrane whose edge is an irregular closed curve. The method is tested by calculating the energy levels for an elliptical and a supercircular boundary and comparing with the results obtained numerically. Further, the phenomenon of level crossing due to shape variation is also discussed.