Two-dimensional Helmholtz equation with zero Dirichlet boundary condition on a circle: Analytic results for boundary deformation, the transition disk-lens

A deformation of a disk D of radius r is described as follows: Let two disks D-1 and D-2 have the same radius r, and let the distance between the two disk centers be 2a, 0 <= a <= r. The deformation transforms D into the intersection D-1 D-2. This deformation is parametrized by epsilon = a/r. For epsilon = 0, there is no deformation, and the deformation starts when epsilon, starting from 0, increases, transforming the disk into a lens. Analytic results are obtained for the eigenvalues of Helmholtz equation with zero Dirichlet boundary condition to the lowest order in epsilon for this deformation. These analytic results are obtained via a Hamiltonian method for solving the Helmholtz equation with zero Dirichlet boundary condition on two intersecting circles of equal radii for 0 <= a <= r. This method involves partial wave expansion and a Green function approach.