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. 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 just used for quantizing RPB's. Next a class of doubly rational polygon billiards (DRPB) is considered for which these real linear relations are rational and their semiclassical quantization by wave function formalism is presented. It is then shown that semiclassical quantization of both the classical momenta and the energy spectra are determined completely by periodic structure of the corresponding RS's. Each RS can be then reduced to elementary polygon patterns (EPP) as its basic periodic elements building it. Each such EPP can be glued to a torus of genus g. Semiclassical wave functions (SWF) are then constructed on EPP's. The SWF's for DRPB's appear to be exact and have forms of coherent sums of plane waves. They satisfy on the billiards boundaries well defined conditions -the Dirichlet, the Neumannn 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 respective incompleteness of SWF's provided by the method used in the paper is discussed. Families of DRPB's can form dens subsets of angle similar rational polygon billiards allowing for approximate semiclassical quantization of the latter. Next general rational polygons are quantized by approximating them by doubly rational ones. A natural extension of the formalism to irrational polygons is described shortly as well. When the semiclassical approximations constructed in the paper appear really as only approximations the latter are controlled by a general criteria of the eigenvalue theory. Finally a relation between the superscar solutions and SWF's constructed in the paper is also discussed. PACS number(s): 03.65.Sq, 02.30.Jr, 02.30.Lt

Superscar states in rational polygon billiards—A reality or an illusion?

“…(b) the linear combinations (2) contains rationals as coefficients independently of the choice of D k , k = 1, 2 (the case of so called doubly rational polygon billiards (DRPB) [8]);…”

High energy semiclassical wave functions in the Bunimovich stadium billiards determined by its periodic orbits

“…Therefore in this way we get with their second property mentioned above. This is because the quantized polygon billiards A ′′ is not of the doubly rational polygon billiards (DRPB) class for which all linear relations on the plane between their independent periods have only rational coefficients [4]. If it was then any choice of allowed EPP for the billiards would be irrelevant for its semiclassical quantization contrary to the considered case which being not DRPB one makes differences in the resulting quantizations because of their dependence on the approximations (6) of irrationals by respective rationals.…”

“…In the previous section the SWFs (12) and (21) were built on aperiodic skeletons giving us the general form of the quantization conditions ( 7), (20) as well as of the SWFs themselves. However the quantization procedure can be performed equally well on periodic skeletons if there are no constraints which can prevent such a quantization [4]. Fortunately in the cases of the considered RPBs the respective constraints can be easily satisfied without any condition on the properties of the polygons.…”

“…Recently however studying the polygon billiards we have shown [3]- [4] that the Maslov approach can be extended also to pseudointegrable system of classical billiards although with some restrictions. Nevertheless still in this extended applications the classical periodic orbits if present seem do not play some distinguished role just providing only examples of specific states of the quantized polygons called superscars [5]- [7].…”

High energy semiclassical wave functions in rational multi-connected and other (Sinai-like) billiards determined by their periodic orbits

“…Billiards due to its extremely simple classical dynamics are particularly convenient dynamical systems for applying the semiclassical approximations to describe their quantum behaviour especially if such approximations are constructing according to the Maslov-Fedoriuk approach [6] where the SWFs are built on classical trajectories. Since in the billiards cases classical trajectories are straight lines broken at the boundaries according to the optical reflection rule the respective SWFs appear to be plane waves with definite wave lengths propagated along these trajectories and reflected by the billiards boundaries according to the same optical rule, see [2] and Sec.2. Therefore the stationary state wave functions arise as a result of interferences of many plane waves reflected in the above way and vanishing by these interferences at billiards boundaries.…”