Schrödinger equation for convex plane polygons: A tiling method for the derivation of eigenvalues and eigenfunctions

Motivated by a recently advanced conjecture on the ergodic properties of Quantum Systems, the problem of solving the Schrodinger equation for a free particle in a plane polygonal enclosure is revisited. It will be shown that two elementary lemmas suffice to give a complete characterization of the polygons for which a solution can be found in terms of a finite superposition of plane waves, without making use of advanced group‐theoretical techniques. It turns out, inter alia, that these polygons, considered as classical billiards, are all and only those which are completely integrable in the sense of Arnold’s theorem.

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