Can billiard eigenstates be approximated by superpositions of plane waves?

Abstract

The plane wave decomposition method (PWDM) is one of the most popular strategies for numerical solution of the quantum billiard problem. The method is based on the assumption that each eigenstate in a billiard can be approximated by a superposition of plane waves at a given energy. By the classical results on the theory of differential operators this can indeed be justified for billiards in convex domains. On the contrary, in the present work we demonstrate that eigenstates of non-convex billiards, in general, cannot be approximated by any solution of the Helmholtz equation regular everywhere in $\R^2$ (in particular, by linear combinations of a finite number of plane waves having the same energy). From this we infer that PWDM cannot be applied to billiards in non-convex domains. Furthermore, it follows from our results that unlike the properties of integrable billiards, where each eigenstate can be extended into the billiard exterior as a regular solution of the Helmholtz equation, the eigenstates of non-convex billiards, in general, do not admit such an extension.