  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. Unfolding rational polygon billiards (RPB)
into corresponding Riemann surfaces (RS) periodic structures of the latter are
demonstrated with 2g independent periods on the respective multitori with g as
their genuses. However it is the two dimensional real space of the real linear
combinations of these periods which is used for quantizing RPB. A class of
doubly rational polygon billiards (DRPB) is distinguished for which these real
linear relations are rational and their semiclassical quantization by wave
function formalism is presented. It is shown that semiclassical quantization of
both the classical momenta and the energy spectra are determined completely by
periodic structure of the corresponding RS. Each RS is then reduced to
elementary polygon patterns (EPP) as its basic periodic elements. Each such EPP
can be glued to a torus of genus g. Semiclassical wave functions (SWF) are then
constructed on EPP. The SWF for DRPB appear to be exact. They satisfy the
Dirichlet, the Neumannn or the mixed boundary conditions. Not every mixing is
allowed however and a respective incompleteness of SWF is discussed. Dens
families of DRPB are used for approximate semiclassical quantization of RPB.
General rational polygons are quantized by approximating them by DRPB. An
extension of the formalism to irrational polygons is described as well. The
semiclassical approximations constructed in the paper are controlled by general
criteria of the eigenvalue theory. A relation between the superscar solutions
and SWF constructed in the paper is also discussed.
