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Eigenvalue Problem in Two Dimension for An Irregular Boundary

An analytical perturbative method is suggested for solving the Helmholtz equation (\bigtriangledown^{2} + k^{2})ψ = 0 in two dimensions where ψ 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.