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
R2 (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.Comment: 23 pages, 5 figure