Semiclassical Wave Functions and Energy Spectra in Polygon Billiards

Abstract

A consistent scheme of semiclassical quantization in polygon billiards by wave function formalism is presented. It is argued that it is in the spirit of the semiclassical wave function formalism to make necessary rationalization of respective quantities accompanied the procedure of the semiclassical quantization in polygon billiards. First the rational polygon billiards and their unfoldings into corresponding Riemann surfaces are considered which show periodic structures with 2g independent periods. The space of the real linear combinations of these periods is however two dimensional and just these space can be rationalized consistently approximating real periods and leading to discrete lattices built of all such approximated periods. It is then shown that semiclassical quantization of both the classical momenta and energy spectra are determined completely by periodic structure of the rationalized lattices. Each Riemann surface can be then reduced to elementary polygon patterns as its basic periodic elements which built it. Each such elementary polygon pattern can be glued to a torus of genus g. Semiclassical wave functions are then constructed on such elementary polygon patterns. These semiclassical wave functions have forms of coherent sums of plane waves and satisfy on the billiards boundaries well defined conditions - the Dirichlet, the Neumann or the mixed ones. Not every mixing of such conditions is allowed however and the respective limitations can ignore some semiclassical states in the presented formalism. A relation between the superscar solutions and the semiclassical wave functions constructed in the paper are discussed. Finally an extension of the formalism on irrational polygons is described shortly as well.