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

Abstract

The rational billiards (RB) are classically pseudointegrable (P.J. Richens, M.V. Berry, Physica D2, 495 (1981), Stefan Giller, arXiv: 1912), i.e. their trajectories in the phase space lie on multi-tori of a genus g defined by 2g independent periods. Each such a multi-torus can be unfolded into elementary polygon pattern (EPP) -a smallest system of mirror images of RB obtained by their consecutive reflections by their sides and containing all different images of RB. A rational billiards Riemann surface (RBRS) corresponding to each RB is then an infinite mosaic made by a periodic distribution of EPP. Periods of RBRS are directly related to periodic orbits of RB. It is shown that any stationary solutions (SS) to the Schrödinger equation (SE) in RB can be extended on the whole RBRS. The extended stationary wave functions (ESS) are then periodic on RBRS with its periods. Conversely, for each system of boundary conditions (i.e. the Dirichlet or the the Neumann ones or their mixture) consistent with EPP one can find so called stationary pre-solutions (SPS) of the Schrödinger equation defined on RBRS and respecting its periodic structure together with their energy spectra. Using SPS one can easily construct SS of RB for most boundary conditions on it by a trivial algebra over SPS. It proves therefore that the energy spectra defined by the boundary conditions for SS corresponding to each RB are totally determined by 2g independent periods of RBRS being homogeneous functions of these periods. RBRS can be considered as a classical construction but in fact it can be done exclusively due to the rationality of the polygon billiards considered. Therefore the approach developed in the present paper can be seen as a new way in obtaining SS to SE in RB. On the other hand our results can be considered also as a generalization on the pseudointegrable systems of the well known semiclassical result corresponding to the integrable rational billiards being however general and exact. SPS can be constructed explicitly for a class of RB which EPP can be decomposed into a set of periodic orbit channel (POC) parallel to each other (POCDRB). For such a class of RB in which POCs find their natural place in the quantization of RB the respective RBRS can be built as a standard multi-sheeted Riemann surface (finitely sheeted in the case of doubly rational billiards (DRPB) and infinitely sheeted in other ones) with a periodic structure. For POCDRB a discussion of the existence of the superscar states (SSS)