Solution of the eigenvalue problem for two-dimensional modulated billiards using a coordinate transformation

We propose a method to solve the eigenvalue problem for a generic two-dimensional modulated billiard by using a coordinate transformation. The transformation allows us to incorporate the effects of the billiard's boundaries into an effective potential, rendering the one particle motion in the two-dimensional billiard equivalent to that of two interacting particles in a one-dimensional geometry. We give the expressions for the Hamiltonian matrix elements for the cases of periodic and closed billiards.