Schrödinger equation for convex plane polygons. II. A no-go theorem for plane waves representation of solutions

In the present article we discuss whether or not ansatzes more general than those considered hitherto, for the representation in terms of plane waves of eigensolutions of the Schrödinger equation for a free particle in a plane convex polygonal domain, lead to the construction of a complete set for a class of domains larger than the well-known one, containing the polygons for which such a complete set can be obtained in terms of a superposition of a finite number of plane waves. Our conclusion is in the negative. Comments are also made on some features of a representation in terms of analytic functions of two complex variables.