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The rational billiards (RB) are classically pseudointegrable, i.e. theirtrajectories in the phase space lie on multi-tori. Each such a multi-torus canbe unfolded into elementary polygon pattern (EPP). A rational billiards Riemannsurface (RBRS) corresponding to each RB is then an infinite mosaic made by aperiodic distribution of EPP. Periods of RBRS are directly related to periodicorbits of RB. It is shown that any stationary solutions (SS) to theSchr\"odinger equation (SE) in RB can be extended on the whole RBRS. Theextended stationary wave functions (ESS) are then periodic on RBRS with itsperiods. Conversely, for each system of boundary conditions (i.e. the Dirichletor the the Neumann ones or their mixture) consistent with EPP one can find socalled stationary pre-solutions (SPS) of the Schr\"odinger equation defined onRBRS and respecting its periodic structure together with their energy spectra.Using SPS one can easily construct SS of RB for most boundary conditions on itby a trivial algebra over SPS. It proves therefore that the energy spectradefined by the boundary conditions for SS corresponding to each RB are totallydetermined by $2g$ independent periods of RBRS being homogeneous functions ofthese periods. RBRS can be constructed exclusively due to the rationality ofthe polygon billiards considered. Therefore the approach developed in thepresent paper can be seen as a new way in obtaining SS to SE in RB. SPS can beconstructed explicitly for a class of RB which EPP can be decomposed into a setof periodic orbit channel (POC) parallel to each other (POCDRB). For such aclass of RB the respective RBRS can be built as a standard multi-sheetedRiemann surface with a periodic structure. For POCDRB a discussion of theexistence of the superscar states (SSS) can be done thoroughly.

Wave Functions and Energy Spectra in Rational Billiards Are Determined Completely by Their Periods