Semiclassical wave functions in billiards built on classical trajectories. Energy quantization, scars and periodic orbits
aa r X i v : . [ m a t h - ph ] A ug Semiclassical wave functions in billiards built onclassical tra jectories. Energy quantization, scarsand periodic orbits
Stefan Giller † and Jaros law Janiak ‡ † Jan D lugosz University in CzestochowaInstitute of PhysicsArmii Krajowej 13/15, 42-200 Czestochowa, Polande-mail: [email protected] ‡ Theoretical Physics Department IIUniversity of L´od´z,Pomorska 149/153, 90-236 L´od´z, Polande-mail: [email protected]
Abstract
A way of construction of semiclassical wave function (SWF) based on the Maslov -Fedoriuk approach is proposed which appears to be appropriate also for systems withchaotic classical limits. Some classical constructions called skeletons are considered. Theskeletons are generalizations of Arnolds’ tori able to gather chaotic dynamics. SWF’s arecontinued by caustic singularities in the configuration space rather then in the phase spaceusing complex time method. The skeleton formulation provides us with a new algorithmfor the semiclassical approximation method which is applied to construct SWF’s as well asto calculate energy spectra for the circular and rectangular billiards as well as to constructthe simplest SWF’s and the respective spectrum for the Bunimovich stadium. The scarphenomena are considered and a possibility of their description by the skeleton methodis discussed.PACS number(s): 03.65.-w, 03.65.Sq, 02.30.Jr, 02.30.Lt, 02.30.MvKey Words: Schr¨odinger equation, semiclassical expansion, Lagrange manifolds, classicaltrajectories, chaotic dynamics, quantum chaos, scars
Introduction
The semiclassical approximation is widely used and by this it is the well known method ofapproximation in quantum physics. There are two basic formulations of the method - the onebased on the wave function formulation of quantum mechanics [1, 6] and the other - on theFeynman paths integral [2, 3]. While both the formulations of quantum mechanics are knownto be equivalent the general wave function formulation of the semiclassical approximation [6]is considered to be not applicable in higher dimensional quantum problems which classicallimits are chaotic.In fact it is a common convince that the only way to formulate the method in the lastcases is the Gutzwiller approach based on the Feynman paths integral [4]. This convincefollows from also a common believe that the wave function formalism can be applied only inthe cases when the classical limit of the quantum problem is the integrable one i.e. if theclassical motion is set on the Arnold tori [5, 7] on which the semiclassical wave functions(SWF) are constructed. As the main argument for such a believe the KAM theorem [5, 7] isinvoked which claims that the Arnold tori structure of the classical phase space disappearsif classical systems become non-integrable. It is argued that because of that the only phasespace finite structure which still survive in chaotic motion of the classical system are periodicorbits which therefore provide us with a skeleton on which the Gutzwiller formula is built.However one can criticise the point of view that the existence of the Arnold tori structureof the phase space is necessary for a possibility to construct the semiclassical wave function,i.e. that Arnold’s tori provide us with a unique support for such a construction. Such adefinite conclusion does certainly not follow from the original local approach of Maslov andFedoriuk to construct SWF’s [6].On the other hand while the results provided by the Gutzwiller method are very rich andappreciated, particularly when energy spectra of the chaotic systems are considered (see forexample [8]), the method itself does not allow us for constructing and discussing propertiesof the wave functions involved in the problems considered. In fact the wave functions arefound in such cases by different methods mostly numerically. It is just due to such numericalcalculations of the wave functions [9, 10] that a phenomenon of scars has been discovered [14]which existence in the wave function patterns is still waiting for its full explanation [16, 19, 15](see also other papers cited in [16]).Billiards while a non-analytic motion area are well known however as examples of the non-integrable two dimensional systems except the known cases of the elliptical and rectangularbilliards. They are widely considered as a simple field of experimental [17, 18] as well astheoretical [20, 21, 22, 23] (and papers cited there) and computational investigations [9,11, 12, 13] allowing to apply many different methods (see Sarnak’s lecture [24] and [25] ofthe same author for an extensive review of the respective theoretical methods covering alsobilliards manifolds).In this paper we are going to develop the SWF formalism which can be applied at leastin principle to non-integrable cases of the two dimensional motions in billiards and which canbe easily extended to higher dimensions.Essentially our approach is initially very close to the one of Maslov and Fedoriuk [6]. Themain difference between them is in a treatment of crossing the singular points of the SWF’sset on caustics. Namely, instead of making the canonical phase space variable transformationsaccompanied by the Fourier transformations of the SWF’s to move through the caustic pointswe apply the analytical continuation on the complex time plane to both the SWF’s and the1lassical trajectories. This greatly simplifies the corresponding procedure in comparison withthe Maslov and Fedoriuk treatment. It is the exceptional role played by the time variable inthe semiclassical limit of the Schr¨odinger equation which permits us for such simplification.Because of this the SWF can be considered as depending effectively on the time variable onlywhile the remaining variables plays the role of spectators.Therefore the SWF’s are first constructed locally on so called bundles of rays to satisfyvanishing boundary conditions. Next they are matching to get a global semiclassical solution.This is done however with the help of an earlier constructed set of reversible in time closedconnected ray bundles called bundle skeleton which play a role of Arnold’s tori except thata number of ray bundles in the skeleton can be infinite.This is just the notion of the ray bundles which allows us to catch a possible chaotic motionin the billiards not to resign from considerations of a set of trajectories on which SWF’s canbe defined while the bundle skeleton idea allows us to close the matching procedure of SWF’sconstructed locally.The paper is organized as follows.In the next section the Maslov - Fedoriuk method of the semiclassical wave functionconstruction is reminded and discussed.In sec.3 the semiclassical wave function is considered as the classical objects which timeevolution is described by the classical equations of motion.In sec.4 the construction of global SWF’s in billiards is given.In sec.5 the circular billiards is considered to demonstrate how the method works in thecase of the presence of caustic.In sec.6 the rectangular billiards is considered as the case deprived of a caustic.In sec.7 our method is applied to the Bunimovich stadium to show its usefulness indescribing the so called bounces ball modes.In sec.8 we discuss a possibility to describe by the skeleton idea the scar phenomenonconsidering such a scar formed around the horizontal periodic orbit in the Bunimovich sta-dium.In sec.9 the results of the paper are summarized and some limitations of the method arediscussed.There are four appendixes attached to the paper which justify the main assumptions usedin the construction of the global SWF’s on skeletons. n -D stationarySchr¨odinger equation Consider the n -dimensional stationary Schr¨odinger equation: △ Ψ( r ) + λ m ¯ h ( E − V ( r ))Ψ( r ) = 0 (1)with a potential V ( r ) , r ∈ R n confining a point particle with a mass m and containing aformal dimensionless parameter λ >
0. For a convenience we shall put further ¯ h = 1 and m = 1. The Schr¨odinger equation is recovered by putting λ = 1 in (1).We would like to construct a solution to Eq.(1) using the idea of Maslov et al [6] andconsidering the wave function Ψ( r ) as defined on families of classical trajectories a dynamicof which is given by the classical Hamiltonian H = p + V ( r ) and which carry an energy E all. 2uch families are constructed locally in the following way. In R n we choose a n − n − on which the initial momenta p ( r ) , r ∈ Σ n − , will be defined so that the pair( r , p ( r )) , r ∈ Σ n − will serve as initial data for the trajectory r ( t ) = f ( r , p ( r ); t ). Thenthe transformation: r → ( t, r ) → ( t, s , ..., s n − ) , r ≡ ( x , ..., x n ) , r ≡ ( x , ( s , ..., s n − ) , ...,x ,n ( s , ..., s n − )), (( s , ..., s n − ) parameterize the hypersurface Σ n − ) is one-to-one up to acaustic surface C n − on which the Jacobean ( f ( r , p ( r ); t ) ≡ ¯ f ( t, s , ..., s n − )): J ( t, s , ..., s n − ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ ¯ f i ∂t , ∂ ¯ f i ∂s j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2)vanishes.A n -dimensional domain Λ n of 2 n -dimensional phase space R n made in this way by thehypersurface Σ n − and trajectories emerging from it is known as the Lagrange manifold [5].Therefore in the variables t, s , ..., s n − the new wave function χ ( t, s , ..., s n − ) satisfiesthe following relation with the previous one: | χ ( t, s , ..., s n − ) | = | Ψ(¯ f ( t, s , ..., s n − ) | | J ( t, s , ..., s n − ) | (3)The particle momentum p on the trajectories r ( t ) = ¯ f ( t, s , ..., s n − ) satisfies of coursethe equation: ∂ ¯ f ( t, s , ..., s n − ) ∂t = p (¯ f ( t, s , ..., s n − )) (4)defining also the Jacobean evolution. Namely: ∂∂t ∂ ¯ f i ( t, s , ..., s n − ) ∂t = n X j =1 ∂p i ∂x j ∂ ¯ f j ( t, s , ..., s n − ) ∂t∂∂t ∂ ¯ f i ( t, s , ..., s n − ) ∂s l = n X j =1 ∂p i ∂x j ∂ ¯ f j ( t, s , ..., s n − ) ∂s l l = 1 , ..., n − ∂J ( t, s , ..., s n − ) ∂t = J ( t, s , ..., s n − ) ∇ p (¯ f ( t, s , ..., s n − )) (6)The above equation is just the Liouville theorem with the solution: J ( t, s , ..., s n − ) = J ( s , ..., s n − ) e R t ∇ p (¯ f ( t ′ ,s ,...,s n − )) dt ′ (7)where J ( s , ..., s n − ) is the value of the Jacobean on the hypersurface Σ n − .It is well known from the classical Hamiltonian mechanics [5] that the action integral: S ( r , r ) = Z rr p ( r ′ ) d r ′ (8)taken on the Lagrange manifold Λ n is a point function of r and r . Therefore taking r as a definite fixed point of the hypersurface Σ n − and denoting by S ( r ) the action function3orresponding to this case we can complete a definition of the wave function χ ( t, s , ..., s n − )by the following equation:Ψ σ (¯ f ( t, s , ..., s n − )) = J − ( t, s , ..., s n − )) e σλiS (¯ f ( t,s ,...,s n − ) χ σ ( t, s , ..., s n − ) (9)where σ = ± is a signature of Ψ σ ( r ).Therefore the quantities involved in the above definitions satisfy the following equations: p ( r ) = ∇ S ( r )12 p ( r ) + V ( r ) − E = 0 △ ( J − χ σ ( r )) + σ iλJ − ( r ) ∇ χ σ ( r ) · p ( r ) + 2 λ ( E − E ) J − ( r ) χ σ ( r ) = 0 r = ¯ f ( t, s , ..., s n − ) (10)By the variables t, s , ..., s n − the third of the last equations can be rewritten in thefollowing form: σ iλ ∂χ σ ( t, s , ..., s n − , λ ) ∂t + J △ (cid:16) J − χ σ ( t, s , ..., s n − , λ ) (cid:17) + λ ( E − E ) χ σ ( t, s , ..., s n − , λ ) = 0 (11)where a dependence of χ σ ( t, s , ..., s n − , λ ) on λ was shown explicitly.The Eq.(11) describes the time evolution of χ σ ( t, s , ..., s n − , λ ) along trajectories startingon the hypersurface Σ n − if its ”initial” values on this surface, i.e. χ σ (0 , s , ..., s n − , λ ) ≡ χ σ ( s , ..., s n − , λ ) are given.We are going to consider the equation (11) in the semiclassical limit λ → + ∞ looking forits solutions in the form of the following asymptotic series: E − E = X k ≥ E k λ − k − χ σ ( t, s , ..., s n − , λ ) = X k ≥ χ σk ( t, s , ..., s n − ) λ − k χ σ ( s , ..., s n − , λ ) = X k ≥ χ σk ( s , ..., s n − ) λ − k (12)Putting λ = 1 in (1), (9) and (12) we get approximate semiclassical solutions to the energyeigenvalue problem of the Schr¨odinger equation.It is to be noticed that for the selfconsistency reasons the semiclassical series for theenergy parameter in (12) starts from the second power of λ − , i.e. this ensures the properhierarchy of steps in the algorithm of semiclassical calculations by which the higher orderterms of the series in (12) are determined by the lower order ones.It should be noticed also that despite the fact that E enters the classical equation ofmotion (10) it is still quantum, i.e. its value depends on ¯ h which is considered to have thedefinite numerical value, i.e. ¯ h is not a parameter. In particular the series (12) representthe inverse power hierarchy in the formal parameter λ , i.e. not in powers of ¯ h , betweensubsequent terms. 4oreover E if quantized can depend on λ . However, whatever is this dependence thesemiclassical series of the difference E − E must be given by (12).Needless to say the introducing λ makes a treatment of the Schr¨odinger equation equiv-alent of course to considering it in the limit ¯ h →
0, i.e. semiclassically, clearly howeverseparating the role of ¯ h as a parameter from its role defining the microscale of quantumphenomena.Substituting (12) into (11) we get: ∂χ σ ( t, s , ..., s n − ) ∂t = 0 ∂χ σk +1 ( t, s , ..., s n − ) ∂t = σi J △ (cid:16) J − χ σk ( t, s , ..., s n − ) (cid:17) + 2 k X l =0 E k − l +1 χ σl ( t, s , ..., s n − ) ! k = 0 , , , ..., (13)with the obvious solutions: χ σ ( t, s , ..., s n − ) ≡ χ σ ( s , ..., s n − ) χ σk +1 ( t, s , ..., s n − ) = χ σk +1 ( s , ..., s n − ) + σi Z t J △ (cid:16) J − χ σk ( t ′ , s , ..., s n − ) (cid:17) + 2 k X l =0 E k − l +1 χ σl ( t ′ , s , ..., s n − ) ! dt ′ k = 0 , , , ..., (14) The recurrent system of equations (13) can be considered also from the classical pointof view as defining the time evolutions of χ σk ( t, s , ..., s n − ) , k = 0 , , , ..., along the classicaltrajectories r ( t ) = ¯ f ( t, s , ..., s n − ). Namely, for each given initial point r ( s , ..., s n − ) anda given trajectory emerging from it let d denotes a distance measured along the trajectoryfrom the point r ( s , ..., s n − ) to the point ( x , ..., x n ) lying on this trajectory. The set( d, s , ..., s n − ) can be used as the new coordinates instead of ( x , ..., x n ). Their mutualrelations are given by: d = ¯ d ( t, s , ..., s n − ) = Z t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ ¯ f ( t ′ , s , ..., s n − ) ∂t ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt ′ (15)and r = ( x , ..., x n ) = ¯ f (¯ t ( d, s , ..., s n − ) , s , ..., s n − ) ≡ g ( d, s , ..., s n − ) (16)where t = ¯ t ( d, s , ..., s n − ) is the solution of d = ¯ d ( t, s , ..., s n − ) with respect to t .Considered on the Lagrange manifold Λ n the system is then governed by the Hamiltonian:¯ H Λ n = 12 p d (cid:12)(cid:12)(cid:12) ∂ g ( d,s ,...,s n − ) ∂d (cid:12)(cid:12)(cid:12) + ¯ V ( d, s , ..., s n − ) (17)5here ¯ V ( d, s , ..., s n − ) ≡ V (¯ f (¯ t ( d, s , ..., s n − ) , s , ..., s n − )) and p d is the momentum conju-gated with d .The corresponding classical equation of motion for χ σ ( t, s , ..., s n − , λ ) on the manifoldΛ n is therefore: dχ σ ( t, s , ..., s n − , λ ) dt = { χ σ , ¯ H Λ n } + ∂χ σ ( t, s , ..., s n − , λ ) ∂t = ∂χ σ ( t, s , ..., s n − , λ ) ∂t (18)since χ σ ( t, s , ..., s n − , λ ) being independent of d commutes with the hamiltonian ¯ H Λ n .Therefore, from (12) and (13) we get on the Lagrange manifold Λ n : dχ σ ( t, s , ..., s n − ) dt = 0 dχ σk +1 ( t, s , ..., s n − ) dt = σi J △ (cid:16) J − χ σk ( t, s , ..., s n − ) (cid:17) + σi k X l =0 E k − l +1 χ σl ( t, s , ..., s n − ) k = 0 , , , ..., (19)Of course (14) provides the solution to the equations (19).To conclude χ σ ( t, s , ..., s n − ) is constant on the Lagrange manifold Λ n according to (14)being obviously independent also of d .It is to be noticed that defining on Σ n − another Lagrange manifold Λ ′ n by choosing a dif-ferent set of trajectories we obtain also another SWF defined on this new Lagrange manifold,i.e. different from the one defined on Λ n . This is why the corresponding χ σ ( t, s , ..., s n − )cannot be considered as the global integral of motion, i.e. in the whole phace space R n ofthe Hamiltonian H - each particular SWF is defined only on a particular Lagrange manifoldΛ n corresponding to it.For future applications it is worth to note that if (11) is obviously invariant on a reparametri-zation of the hypersurface Σ n − it is also invariant on the following change of variables: t → τ ( s , ..., s n − ) − ts k → h k ( s , ..., s n − ) k = 1 , ..., n − χ σ ( t, s , ..., s n − , λ ) → χ − σ ( t, s , ..., s n − , λ ) ≡ J − ( h − ( s , ..., s n − ) , ..., h − n − ( s , ..., s n − )) × χ σ ( τ ( s , ..., s n − ) − t, h − ( s , ..., s n − ) , ..., h − n − ( s , ..., s n − ) , λ ) (21)where J ( s , ..., s n − ) is the Jacobean of the transformation (20). Let us apply the above formalism to construct continuous semiclassical wave functionsinside billiards B (Fig.1) vanishing on its boundary. Such a construction will be done inseveral steps the first one consisting of some geometrical preliminaries describing a notion ofa skeleton, i.e. the closed set of families of trajectories which forms a base on which SWF’sare constructed. 6igure 1: An arbitrary billiards We shall assume that the billiards is classical, i.e. its boundary ∂B is a closed curveindependent of λ and given by r = r ( s ) = [ x ( s ) , y ( s )] where s is a distance of a boundarypoint r ( s ) measured clockwise along ∂B from some other point A of ∂B chosen arbitrary, i.e. s ( A ) = 0. Both x ( s ) and y ( s ) are continues. The curve however consists of a finite number q, q ≥ A , ..., A q with respective length L , ..., L q , so that the derivatives x ′ ( s ) and y ′ ( s ) are discontinuous in a finite number of points on the segment 0 ≤ s ≤ L where L = L + · · · + L q is the total length of ∂B . Both x ( s ) and y ( s ) are of course periodicwith the period equal to L . We shall identify the point A with the point beginning the arc A .Next we define a bundle of rays as a family of trajectories in the following way.Let A k ( u, l ) , L + · · · + L k − ≤ u ≤ L + · · · + L k , < l ≤ L k , be an open connectedpiece of the arc A k beginning at s = u and having a length l .Let further r k ( s, t ; u, l ) , u < s < u + l, ≤ t, be a family of trajectories given byangles γ k ( s ; u, l ) , ≤ γ k ( s ; u, l ) ≤ π , at which the trajectories escape from A k ( u, l ). Theangles γ ( s ; u, l ) are smooth functions of s and are measured with respect to the x -axis whilethe tangential vectors t ( s ) = [ dx ( s ) ds , dy ( s ) ds ] = [cos β ( s ) , sin β ( s )] are inclined to the x -axisat angles β ( s ) (Fig.1). The latter angle can be discontinuous at the points where x ′ ( s )and y ′ ( s ) are discontinuous. Then the angle α k ( s ; u, l ) = γ k ( s ; u, l ) − β k ( s ) is made bythe classical ball momentum p ( s ; u, l ) on the trajectory with the tangent vector t ( s ), i.e. p ( s ; u, l ) · t ( s ) = p cos α k ( s ; u, l ). It is assumed that 0 < α k ( s ; u, l ) < π .The classical time evolution of the family r k ( s, t ; u, l ) , u < s < u + l, is therefore thefollowing r k ( s, t ; u, l ) = r ( s ) + p ( s ; u, l ) t, r ( s ) ∈ A k (22)7here p ( s ; u, l ) = [ p cos γ k ( s ; u, l ) , p sin γ k ( s ; u, l )] satisfies the classical equations of motions(10), i.e. p ( s ; u, l ) = 2 E (again we put m = 1 for the billiard ball mass).The trajectories (22) define of course the change of variables ( x, y ) → ( t, s ), in vicinity of A k ( u, l ), i.e. x = f k ( t, s ; u, l ) , y = g k ( t, s ; u, l ) with the Jacobean:˜ J k ( t, s ; u, l ) = p γ ′ k ( s ; u, l ) t − p | t ( s ) | sin α k ( s ; u, l ) = p γ ′ k ( s ; u, l ) t − p sin α k ( s ; u, l ) (23)since | t ( s ) | = 1.The family of trajectories defined in the above way will be called a bundle of rays emerging from the segment A k ( u, l ) of A k and will be denoted by B k ( u, l ) while the trajectoryitself will be called rays .Since each ray of the bundle B k ( u, l ) after some time τ k ( s ; u, l ) , r ( s ) ∈ A k ( u, l ), (differentfor different rays) achieves another point of the boundary ∂B it means that the bundle B k ( u, l )maps the segment A k ( u, l ) into another piece BA k ( u, l ) of the boundary ∂B . In general itis assumed that this map of A k ( u, l ) into BA k ( u, l ) provided by the transformation (22) isone-to-one except the caustic points of BA k ( u, l ) in which ˜ J K ( τ k ( s ; u, l ) , h k ( s ; u, l ); u, l ) =0 , r ( h k ( s ; u, l )) ∈ BA k ( u, l ). Here h k ( s ; u, l ) , r ( s ) ∈ A k ( u, l ), realizes explicitly this map.The caustic points will be assumed to be isolated singular points of the map A k ( u, l ) → BA k ( u, l ). Typically they need a special care if they are.By DB k ( u, l ) will be denoted a domain of the billiards B covered by rays of the bundle B k ( u, l ) emerging from A k ( u, l ) and ending at BA k ( u, l ). The domain DB k ( u, l ) is locally aLagrangian manifold on which each loop integral H p · d r vanishes.Assume further the boundary ∂B to be a mirror-like, i.e. reflecting the incoming raysaccording to the reflection principle of the geometrical optics and let A k ′ ( u ′ , l ′ ) be a piece ofanother arc A k ′ of ∂B such that A k ′ ( u ′ , l ′ ) ∩ BA k ( u, l ) = ⊘ on which another bundle of rays B k ′ ( u ′ , l ′ ) = { r k ′ ( s, t ; u ′ , l ′ ) : u ′ < s < u ′ + l ′ , ≤ t } is defined.If on the segment A k ′ ( u ′ , l ′ ) ∩ BA k ( u, l ) the ray bundle B k ′ ( u ′ , l ′ ) coincides with thereflected one we call the ray bundle B k ′ ( u ′ , l ′ ) a reflection of the bundle B k ( u, l ) on thesegment mentioned.The reflection operation over the bundle B k ( u, l ) will be denoted by Π so that Π B k ( u, l )denotes the set of all rays arising by the reflection of the bundle B k ( u, l ) of ∂B .Consider now a family of the disjoint ray bundles B = S B k ( u, l ) , B k ( u, l ) ∩ B k ′ ( u ′ , l ′ ) = ⊘ if B k ( u, l ) = B k ′ ( u ′ , l ′ ).The family B will be called closed under reflection Π on the boundary ∂B if the followingtwo conditions are satisfied for each bundle B k ( u, l ) ∈ B :Π B k ( u, l ) = n [ j =1 B j ( u j , l j ) ∩ Π B k ( u, l ) , B j ( u j , l j ) ∈ A i j , j = 1 , ..., nB k ( u, l ) = m [ i =1 Π B i ( u i , l i ) ∩ B k ( u, l ) , B i ( u i , l i ) ∈ A j i , i = 1 , ..., m Π B = B (24)A closed family B ′ = S B k ′ ( u ′ , l ′ ) is embedded into a closed family B = S B k ( u, l ) if eachray bundle of B ′ is a subset of some ray bundle of B and each bundle of B contains somebundle of B ′ . 8 closed bundle family is called connected if it cannot be decomposed into a sum ofanother two disjoint closed bundle families.A closed connected bundle family will be called a Lagrange bundle skeleton or simplya skeleton if it cannot be embedded into another closed connected bundle family.From now on all considered bundle families will be assumed to be skeletons.A basic property of each ray belonging to a skeleton B is that by its time evolution andbounces on the billiards boundary it will never leave B .Another basic property of a skeleton B is its completeness, i.e. none bundle can be addedto or removed from the skeleton not destroying it.Let us now reverse in time all trajectories belonging to B . This operation leads us againto some skeleton B A which will be called associated with B .Bundles of B A are obtained simply from the corresponding bundles of B . Namely witheach bundle B k ( u, l ) of B let us associate a bundle B Ak ( u, l ) which trajectories satisfy thefollowing condition: γ Ak ( s ; u, l ) = π + 2 β ( s ) − γ k ( s ; u, l ) , u < s < u + l (25)i.e. these trajectories are just the reflections on ∂B of the time reversed trajectories definedby γ k ( s ; u, l ) and belonging to B k ( u, l ).The skeleton B A is organized by all bundles B Ak ( u, l ) obtained in the above way. Consider a bundle skeleton B . On each of its ray bundle B k ( u, l ) we can now define thefollowing pair of SWF’s Ψ σk ( t, s ; u, l ; λ ) , σ = ± :Ψ σk ( t, s ; u, l ; λ ) = ˜ J − k ( t, s ; u, l ) e σiλ ( p t + p R s cos α k ( s ; u,l ) ds ′ ) χ σk ( t, s ; u, l ; λ ) (26)where p = 2 E and χ σk ( t, s ; u, l ; λ ) , σ = ± , are given by (12) and (14).Exactly in the same way we can define a pair Ψ σA ; k ( t, s ; u, l ; λ ) , σ = ± , of SWF’s on thecorresponding associated bundle B Ak ( u, l ):Ψ σA ; k ( t, s ; u, l ; λ ) = ˜ J − A ; k ( t, s ; u, l ) e σiλ ( p t − p R s cos α k ( s ; u,l ) ds ′ ) χ σA ; k ( t, s ; u, l ; λ ) (27)It will be also convenient for further considerations to substitute the time variable t bythe distance variable d = pt and consequently to give the trajectories (22), the Jacobean (23)and the wave function (26) the following forms: r k ( d, s ; u, l ) = r ( s ) + d ( s ; u, l ) d ( s ; u, l ) = p t = [ d cos γ k ( s ; u, l ) , d sin γ k ( s ; u, l )] r ( s ) ∈ A k ( u, l ) (28)and J k ( d, s ; u, l ) = 1 p ˜ J k ( t, s ; u, l ) = ∂γ k ( s ; u, l ) ∂s d − sin α k ( s ; u, l ) r ( s ) ∈ A k ( u, l ) (29)9nd Ψ σk ( d, s ; u, l ; λ ) = J − k ( d, s ; u, l ) e σik ( d + R s cos α k ( s ′ ; u,l ) ds ′ ) ¯ χ σk ( d, s ; u, l ; λ ) r ( s ) ∈ A k ( u, l ) (30)where ¯ χ σk ( d, s ; u, l ; λ ) ≡ pχ σk ( dp , s ; u, l ; λ ) , σ = ± , and k = pλ is the wave number of thebilliards ball.Nevertheless, for simplicity of notations, the bar over ¯ χ σk ( d, s ; u, l ; λ ) will be dropped inour further considerations.By the variable d the solutions (14) can be rewritten in the form: χ σk, ( d, s ; u, l ) ≡ χ σk, ( s ; u, l ) χ σk,j +1 ( d, s ; u, l ) = χ σk,j +1 ( s ; u, l ) + σi p Z d D k ( a, s ; u, l ) χ σk,j ( a, s ; u, l ) + 2 j X m =0 E j − m +1 χ σk,m ( a, s ; u, l ) daj = 0 , , , ..., (31)where D k ( d, s ; u, l ) = J k ( d, s ; u, l ) ·△ k ( d, s ; u, l ) · J − k ( d, s ; u, l ) and △ k ( d, s ; u, l ) is the Laplaceanexpressed by the variables d and s corresponding to the B k ( u, l )-bundle.Ψ σk ( d, s ; u, l ; λ ) , σ = ± , are defined initially in the domain D k ( u, l ) of the billiards whichboundary ∂D k ( u, l ) contains of course A k ( u, l ). A pattern of the remaining part of ∂D k ( u, l )depends on the corresponding caustic K k ( u, l ) = { ( f k ( K k ( s ; u, l ) , s ; u, l ) , g k ( K k ( s ; u, l ) , s ; u, l )) : J k ( K k ( s ; u, l ) , s ; u, l ) = 0 , r ( s ) ∈ A k ( u, l ) } . It follows from (23) that K k ( s ; u, l ) = sin α k ( s ; u, l ) ∂γ k ( s ; u,l ) ∂s , r ( s ) ∈ A k ( u, l ) (32)and K k ( s ; u, l ) can escape to infinity if ∂γ k ( s ; u,l ) ∂s → K k ( u, l ) ∩ B is not empty then it belongs to ∂D k ( u, l ). The remaining rays of B k ( u, l ) which are not tangent to K k ( u, l ) ∩ B end at BA k ( u, l ) and complete in this way thedomain D k ( u, l ).An important property of the representation (26) is its uniqueness, i.e. for two differentbundles defined on the segment A k ( u, l ) this representation provides us with two differentpairs of Ψ σk ( d, s ; u, l ; λ ) , σ = ± . This conclusion follows from the fact that for λ sufficientlylarge the SWF’s are determined only by exponentials and the latter are different at the samepoints ( x, y ) for different bundles. Ψ σk ( d, s ; u, l ; λ ) through a skeleton Ψ σk ( d, s ; u, l ; λ ) , σ = ± , being defined in D k ( u, l ) ∈ B k ( u, l ) can be continued on the wholeskeleton by accepting two rules the solutions have to satisfy.A need for the first rule to be formulated arises when the solutions propagate by thecaustic where they are singular. These are branch points singularities on the d -plane of thetype ( d − d c ( s )) − where J k ( d c ( s ) , s ; u, l ) = 0. These points have to be avoided somehowwhen continuing the solutions along the rays of B k ( u, l ). Relying on the results of App.Bwe accept the rule of avoiding these points clockwise by Ψ + k ( d, s ; u, l ; λ ) and above them andanticlockwise by Ψ − k ( d, s ; u, l ; λ ) and below them.10he second rule has to be given to describe bouncings of Ψ σk ( d, s ; u, l ; λ ) , σ = ± , by thebilliards boundary.Let B k ′ ( u ′ , l ′ ) be a reflection of the bundle B k ( u, l ) i.e. B k ′ ( u ′ , l ′ ) ⊂ Π B k ( u, l ). Continua-tions of Ψ σk ( d, s ; u, l ; λ ) , σ = ± , onto B k ′ ( u ′ , l ′ ) is given by the following formula:Ψ σ,contk ( d, s ; u ′ , l ′ ; λ ) = − η σ J − k ′ ( d, s ; u ′ , l ′ ) × e σiλp ( d − R su ′ cos α k ′ ( s ′ ; u ′ ,l ′ ) ds ′ + δ k ( u,l ) ) χ σ,contk ( d, s ; u ′ , l ′ ; λ ) δ k ( u, l ) = D ( h − k ( s ; u, l ); u, l ) + Z su ′ cos α k ′ ( s ′ ; u ′ , l ′ ) ds ′ − Z h − k ( s ; u,l ) u cos α k ( s ′ ; u, l ) ds ′ χ σ,contk (0 , s ; u ′ , l ′ ; λ ) = (cid:12)(cid:12)(cid:12)(cid:12) ∂h k ( s ′ ; u, l ) ∂s ′ (cid:12)(cid:12)(cid:12)(cid:12) − s ′ = h − k ( s ; u,l ) χ σk ( D ( h − k ( s ; u, l ); u, l ) , h − k ( s ; u, l ); u, l ; λ ) r ( s ) = r ( h − k ( s ; u, l )) + D ( h − k ( s ; u, l ); u, l ) (33)The factor η σ in (33) depends on whether there was a caustic point on the way of contin-uations of Ψ σk ( d, s ; u, l ; λ ) in which case η σ = σi but η σ ≡ D ( s ; u, l ) = | D ( s ; u, l ) | denotes the distance between the points r ( h k ( s ; u, l )) and r ( s )of the boundary ∂B .It is to be noticed that δ k ( u, l ) is s -independent (see App.C). χ σ,contk ( d, s ; u ′ , l ′ ; λ ) is defined by (31) with the initial values given by the correspondingsemiclassical series of χ σ,contk (0 , s ; u ′ , l ′ ; λ ) as given by (33). Due to the chain property ofthe formulae (31) such a definition of χ σ,contk ( d, s ; u ′ , l ′ ; λ ) ensures that χ σk ( d, s ; u, l ; λ ) can beconsidered as a continuous function of d on the skeleton as d changes along rays arising bysubsequent reflections of the one starting from the bundle B k ( u, l ).Another property of the definition (33) is that close to the arc A k ′ there is a domainin which both the SWF’s Ψ σ,contk ( d ′ , s ′ ; u ′ , l ′ ; λ ) and Ψ σk ( d, s ; u, l ; λ ) are defined so that thesuperposition: Ψ σ,supk ( x, y, λ ) ≡ Ψ σ,contk ( d ′ , s ′ ; u ′ , l ′ ; λ ) + Ψ σk ( d, s ; u, l ; λ ) x = x ( s ) + d cos α k ( s ; u, l ) = x ( s ′ ) + d ′ cos α k ′ ( s ′ ; u ′ , l ′ ) y = y ( s ) + d sin α k ( s ; u, l ) = y ( s ′ ) + d ′ sin α k ′ ( s ′ ; u ′ , l ′ ) (34)can be done.It follows easily from (33) that on the arc A k ′ , i.e. for d ′ = 0 this superposition vanishes. Another obvious property of Ψ σk ( d, s ; u, l ; λ ) , σ = ± , is that they cannot vanish on A k ( u, l )unless χ σk ( d, s ; u, l ; λ ) vanish there identically. Therefore a wave function Ψ as ; σk ( x, y ; u, l ; λ )vanishing on A k ( u, l ) should be represented in the semiclassical limit by a linear combinationof at least two SWF’s of the form (26). The superposition (34) is an important example ofsuch a combination. It is shown however in App.A on a more general level of considerationsthat the proper linear combinations have to be the following:Ψ as ; σk ( x, y ; u, l ; λ ) = Ψ σk ( d , s ; u, l ; λ ) + Ψ − σA ; k ( d , s ; u, l ; λ ) = J − k ( d , s ; u, l ) e iσk ( d + R s cos α k ( s ′ ; u,l ) ds ′ ) χ σk ( d , s ; u, l ; λ ) + J − A ; k ( d , s ; u, l ) e − iσk ( d − R s cos α k ( s ′ ; u,l ) ds ′ ) χ − σA ; k ( d , s ; u, l ; λ ) (35)11ith the following boundary conditions: χ σk (0 , s ; u, l ; λ ) + χ − σA ; k (0 , s ; u, l ; λ ) = 0 r ( s ) ∈ A k ( u, l ) (36)while the point ( x, y ) is the cross point of the respective trajectories belonging to differentbundles, i.e. r ≡ [ x, y ] = r k ( d , s ; u, l ) = r ( s ) + d ( s ; u, l ) = r A ; k ( d , s ; u, l ) = r ( s ) + d ( s ; u, l ) r k ( d, s ; u, l ) ∈ B k ( u, l ) , r A ; k ( d, s ; u, l ) ∈ B Ak ( u, l ) (37)The vanishing superposition (35) if defined on the bundle B k ′ ( u ′ , l ′ ) suggests thatΨ − σA ; k ′ ( d, s ; u ′ , l ′ ; λ ) should be related somehow to the SWF Ψ σk ( d, h − k ( s ; u, l ); u, l ); u, l ; λ ) de-fined on the bundle B k ( u, l ) which the previous one is a reflection. In the next section thisrelation is established as a condition matching both the solutions. Let now Ψ as ; σk ( x, y ; u, l ; λ )’s be defined in the above way on the corresponding bun-dles of B and B A . Suppose that the corresponding SWF’s Ψ σk ( d, s ; u, l ; λ ) , σ = ± , andΨ A ; σk ( d, s ; u, l ; λ ) , σ = ± , which construct them can be continued along the respective bun-dles on which they are defined. Note that the domains D k ( u, l ) and D Ak ( u, l ) where theseSWF’s are initially defined are then extended to DB k ( u, l ) and DB Ak ( u, l ) respectively. Weare then faced with the problem of matching the continued solutions with the others definedon the same skeletons B and B A .Anticipating the results of App.B such a matching should be done as follows.Let B k ( u, l ) be a reflection of the bundles B k ( u , l ) , B k ( u , l ) , . . . , B k n ( u n , l n ) satis-fying (24). Let Ψ σk ( d, s ; u , l ; λ ) , Ψ σk ( d, s ; u , l ; λ ) , . . . , Ψ σk n ( d, s ; u n , l n ; λ ) be defined andcontinued on the respective bundles B k ( u , l ) , B k ( u , l ) , . . . , B k n ( u n , l n ). Let furtherΨ − σA ; k ( d, s ; u, l ; λ ) be defined on the bundle B Ak ( u, l ) while Ψ σk ( d, s ; u, l ; λ ) on the bundle B k ( u, l )being both related by the boundary condition (36).Then we make the following identification:Ψ − σA ; k ( d, h ( s ; u j , l j ); u, l ; λ ) = Ψ σk j ( D ( s ; u j , l j ) − d, s ; u j , l j ; λ ) r ( h ( s ; u j , l j )) ∈ A k ( u, l ) ∩ BL ( u j , l j ) r ( s ) ∈ L ( u j , l j ) r ( h ( s ; u j , l j )) = r ( s ) + D ( s ; u j , l j ) , j = 1 , ...n (38)Rewritten in terms of the χ -coefficients eq.(38) gives: χ − σA ; k ( d, h ( s ; u j , l j ); u, l ; λ ) = η σ e σiλpδ ( u j ,l j ) (cid:12)(cid:12)(cid:12)(cid:12) ∂h ( s ; u j , l j ) ∂s (cid:12)(cid:12)(cid:12)(cid:12) − χ σk j ( D ( s ; u j , l j ) − d, s ; u j , l j ; λ ) δ ( u j , l j ) = D ( s ; u j , l j ) + Z su j cos α k j ( s ′ ; u j , l j ) ds ′ − Z h ( s ; u j ,l j ) u cos α k ( s ′ ; u, l ) ds ′ r ( h ( s ; u j , l j )) ∈ A k ( u, l ) ∩ BL ( u j , l j ) r ( s ) ∈ L ( u j , l j ) r ( h ( s ; u j , l j )) = r ( s ) + D ( s ; u j , l j ) , j = 1 , ...n (39)12s previously δ ( u j , l j ) in the above formula is s -independent (see App.C). Due to thatand due to the properties (20) and (21) the rhs of (39) satisfies (11) as it should.Putting d = 0 in (39) and taking into account (36) we get: χ σk (0 , h ( s ; u j , l j ); u, l ; λ ) = − χ − σA ; k (0 , h ( s ; u j , l j ); u, l ; λ ) = − η σ e σiλpδ ( u j ,l j ) (cid:12)(cid:12)(cid:12)(cid:12) ∂h ( s ; u j , l j ) ∂s (cid:12)(cid:12)(cid:12)(cid:12) − χ σk j ( D ( s ; u j , l j ) , s ; u j , l j ; λ ) = − η σ e σiλpδ ( u j ,l j ) χ σ,contk j (0 , h ( s ; u j , l j ); u, l ; λ ) r ( h ( s ; u j , l j )) ∈ A k ( u, l ) ∩ BL ( u j , l j ) r ( s ) ∈ L ( u j , l j ) r ( h ( s ; u j , l j )) = r ( s ) + D ( s ; u j , l j ) , j = 1 , ...n (40)Comparing (40) with (33) it is easily seen that Ψ σk ( d, s ; u, l ; λ ) has to coincide withΨ σ,contk j ( d, s ; u, l ; λ ). Let r = ( x, y ) be a fixed point of the billiards. Let D ( x, y ) denote a set of all D ( u, l ), B ( u, l ) ∈ B which contain this point and D A ( x, y ) the respective set of D A ( u, l ) , B A ( u, l ) ∈ B A . Let further DB ( x, y ) denote a set of all DB ( u, l ), B ( u, l ) ∈ B which contain the point( x, y ) and DB A ( x, y ) the respective set of DB A ( u, l ) , B A ( u, l ) ∈ B A .SWF’s Ψ as ; σ B ( x, y, λ ) vanishing on the billiards boundary can now be defined on B and B A as follows: Ψ as ; σ B ( x, y, λ ) = X D ( u,l ) ∈ D ( x,y ) Ψ σ ( d ( u, l ) , s ( u, l ); u, l ; λ ) + X D A ( u,l ) ∈ D A ( x,y ) Ψ − σA ( d ( u, l ) , s ( u, l ); u, l ; λ ) = X DB ( u,l ) ∈ DB ( x,y ) Ψ σ ( d ( u, l ) , s ( u, l ); u, l ; λ ) = X DB A ( u,l ) ∈ DB A ( x,y ) Ψ − σA ( d ( u, l ) , s ( u, l ); u, l ; λ ) r = r ( s ( u, l )) + d ( u, l ) (41)The SWF’s Ψ σ ( d, s ; u, l ; λ ) defined on bundles of B and Ψ σA ( d, s ; u, l ; λ ) defined on re-spective bundles of B A are related with each other by the boundary conditions (36) and bymatching conditions (38)-(40).The solutions Ψ as ;+ B ( x, y, λ ) and Ψ as ; − B ( x, y, λ ) coincide if and only if B = B A .It is clear that the conditions (40) have to determine also the χ -factors χ σk ( s ; u, l ; λ )for all the bundles B k ( u, l ) which are the ”initial” conditions for both χ σk ( d, s ; u, l ; λ ) and χ σA ; k ( d, s ; u, l ; λ ), i.e. χ σk ( s ; u, l ; λ ) ≡ χ σk (0 , s ; u, l ; λ ) ≡ − χ σA ; k (0 , s ; u, l ; λ ). Nevertheless theseconditions cannot be given arbitrarily. Just opposite all χ k ( s ; u, l ; λ ) have to satisfy (40) inselfconsistent way. It is not easy to predict how this can be realized.Certainly one has to take into account that χ σk ( s ; u, l ; λ ) being given at the point r ( s ( u, l ))of the bundle B k ( u, l ) are distributed along the boundary ∂B by subsequent reflections ofthe initial ray r = r ( s ( u, l )) + d ( u, l ) which carries on χ σk ( d, s ; u, l ; λ ) according the for-mulae (31), (33) and (40). However the ray come back eventually to the segment A k ( u, l )defining χ σk ( s ; u, l ; λ ) in some new point r ( s ′ ( u, l )) of this segment. Naturally a value of χ σ,contk ( D ( u, l ) , s ′ ( u, l ); u ′ , l ′ ; λ ) at this point has to coincide with χ σk ( s ′ ( u, l ); u, l ; λ ) defined13lready at this point. This is just one of the selfconsistency conditions contained in (40) (see”the last quantization condition” (49) below).It seems difficult however to find a general systematic way of exploring the system (40)to get solutions satisfying it. Nevertheless one can always think at least minimally trying toguess reasonable solutions to this system. In the next sections we will try to use both theapproaches for some particular cases of billiards.The formulae (39) and (40) defines the conditions which the SWF’s χ σk ( d, h ( s ; u j , l j ); u, l ; λ )should satisfy when bouncing from the billiards boundary. Nevertheless this condition shouldbe specified additionally with respect to its factors. Namely, taking their large λ -limit weget: χ σk, ( h ( s ; u j , l j ); u, l ) = − δ (cid:12)(cid:12)(cid:12)(cid:12) ∂h ( s ; u j , l j ) ∂s (cid:12)(cid:12)(cid:12)(cid:12) − e σiλpδ ( u j ,l j ) χ σk j , ( s ; u j , l j ) χ σk,r +1 ( h ( s ; u j , l j ); u, l ) = − δ (cid:12)(cid:12)(cid:12)(cid:12) ∂h ( s ; u j , l j ) ∂s (cid:12)(cid:12)(cid:12)(cid:12) − e σiλpδ ( u j ,l j ) × χ σk j ,r +1 ( s ; u j , l j ) + σi p Z D ( s ; u j ,l j )0 (cid:18) J △ (cid:16) J − χ σk j ,r ( a, s ; u j , l j ) (cid:17) +2 r X l =0 E r − l +1 χ σk j ,l ( a, s ; u j , l j ) ! da ! r = 0 , , , ..., (42)The above equations should be satisfied on each bundle B k ( u, l ) of the skeleton B .The first of the equations (42) should determine the classical quantities, namely theskeleton B and the ”classical” energy E = p and by them define the JWKB approximationof the SWF’s. Namely:Ψ JW KB ; σ B ( x, y, λ ) = X DB ( u,l ) ∈ D ( x,y ) Ψ JW KB ; σ ( d ( u, l ) , s ( u, l ); u, l ) = X DB ( u,l ) ∈ D ( x,y ) J − ( d ( u, l ) , s ( u, l )) e iλp (cid:16) d ( u,l )+ R s ( u,l ) u cos α ( s ′ ; u,l ) ds ′ (cid:17) χ σ ( d ( u, l ) , s ( u, l ); u, l ) (43)The remaining equations determine quantum corrections to the ”classical” ones involvedin (43).However it is easy to note that for the selfconsistency of the equations (42) it is necessaryfor the exponent e σiλpδ ( u j ,l j ) to be independent of λ , i.e. we have to have on each bundle B k ( u, l ) of B : λpδ k ( u, l ) = φ k ( u, l ) B k ( u, l ) ⊂ B (44)where δ k ( u, l ) is given by (33) and φ k ( u, l ) is a λ -independent constant.The equations (44) have to define both the skeleton B and the energy E = p .Taking into account the last conclusions we get the following final set of the recurrentquantization conditions: λpδ k ( u, l ) = φ k ( u, l )14 σk, ( h ( s ; u j , l j ); u, l ) = − δe σiφ kj ( u j ,l j ) (cid:12)(cid:12)(cid:12)(cid:12) ∂h ( s ; u j , l j ) ∂s (cid:12)(cid:12)(cid:12)(cid:12) − χ σk j , ( s ; u j , l j ) χ σk,r +1 ( h ( s ; u j , l j ); u, l ) = − δe σiφ kj ( u j ,l j ) (cid:12)(cid:12)(cid:12)(cid:12) ∂h ( s ; u j , l j ) ∂s (cid:12)(cid:12)(cid:12)(cid:12) − (cid:18) χ σk j ,r +1 ( s ; u j , l j )+ σi p Z D ( s ; u j ,l j )0 J △ (cid:16) J − χ σk j ,r ( a, s ; u j , l j ) (cid:17) + 2 r X l =0 E r − l +1 χ σk j ,l ( a, s ; u j , l j ) ! da ! r = 0 , , , ..., (45)together with: χ σk, ( d, s ; u, l ) ≡ χ σk, ( s ; u, l ) χ σk,r +1 ( d, s ; u, l ) = χ σk,r +1 ( s ; u, l ) + σi p Z d J △ (cid:16) J − χ σk,r ( a, s ; u, l ) (cid:17) + 2 r X m =0 E r − l +1 χ σk,m ( a, s ; u, l ) ! dar = 0 , , , ..., (46) A skeleton on which SWF Ψ as ( x, y, λ ) is built in the way described above does not ensureautomatically that this SWF will be defined continuously on it. It is certainly continuousinside each bundle of the skeleton but it can appear to be discontinuous if a point ( x, y )crosses bundle boundaries. In general the following circumstances can accompany such acrossing:1. the crossed boundary of DB k ( u, l ) coincides with a boundary of DB k ′ ( u + l, l ′ ) of aneighboring bundle B k ′ ( u + l, l ′ );2. there is no such respective neighboring bundle.In the first of the above cases the corresponding Ψ ± k ( d, s ; u, l ; λ ) , ( x ( d, s ) , y ( d, s )) ∈ DB k ( u, l ), and Ψ ± k ′ ( d, s ; u + l, l ′ ; λ ) , ( x ( d, s ) , y ( d, s )) ∈ DB k ′ ( u + l, l ′ ), defined by (35) havebe identified on the common boundary, i.e.Ψ ± k ( d, s ; u, l ; λ ) = Ψ ± k ′ ( d, s ; u + l, l ′ ; λ )( x ( d, s ) , y ( d, s )) ∈ ∂DB k ( u, l ) ∩ ∂DB k ′ ( u + l, l ′ ) = ⊘ (47)In the second case however the corresponding SWF’s Ψ ± k ( d, s ; u, l ; λ ) , ( x ( d, s ) , y ( d, s )) ∈ DB k ( u, l ) have to vanish on such a boundary of DB k ( u, l ), i.e.Ψ ± k ( d, s ; u, l ; λ ) = 0( x ( d, s ) , y ( d, s )) ∈ ∂DB k ( u, l ) (48)The last condition though necessary seems to be a little bit arbitrary. However we shouldremember that our calculations are performed in the semiclassical regime, i.e. in the classi-cally allowed regions (bundles) outside which the semiclassical wave functions cannot exist.Physically this means obviously that outside each bundle a corresponding piece of the wavefunction represented on the bundle by its respective semiclassical approximation has to vanishexponentially when moving away from the bundle.15t will appear in the next sections that both the possibilities (47) and (48) will be metand will have to be applied to solve the problem of the semiclassical quantization.By the definition (41) and by the recurrent equations (45)-(46) and by the conditions(47)-(48) we have formulated the quantization procedure which should determine skeletons,the SWF’s Ψ as ; σ B ( x, y, λ ) and the corresponding energy levels. In the next two sections weshall apply this procedure to the simplest well known cases of billiards, i.e. to the circularand the rectangular ones.Let us note finally that if B = B A then energy levels corresponding to the skeleton B haveto be degenerate. This conclusion follows easily from the form of the quantization conditions(45)-(46) and (47)-(48) showing that the complex conjugations of Ψ as ; σ B ( x, y, λ ) satisfy alsothese conditions with the same semiclassical energy E . The two corresponding solutions areof course Ψ as ; ± B ( x, y, λ ). For a given billiards there can be skeletons with a finite number of bundles as well as withan infinite one. The semiclassical quantization procedure described in the previous sectionsseems to be easily applied to the finite bundle number skeletons. Namely in such a casefollowing a trajectory starting from a bundle B k ( u, l ) we have to approach the same bundleafter a finite number of bounces. The corresponding semiclassical wave function propagatedby the skeleton has therefore to come back to its initial form achieving again the initialbundle. This condition closes the process of quantization formulated in the previous sections.The respective conditions are of course the following:( − δ ) n exp X B k ( u,l ) ∈ B φ k ( u, l ) = 1 χ σ,contk ( D, s ; u, l ; λ ) = χ σk ( s ; u, l ; λ ) (49)where n is a number of bounces and D is the total distance passed by the billiards ball alongthe investigated trajectory.The cases of skeletons with an infinite number of bundles are much more difficult forinvestigations. Such skeletons should be typical for chaotic billiards. According to its defini-tion a bundles B k ( u, l ) can bifurcate after the reflection by the billiards boundary into manydifferent subbundles, i.e. parts of other bundles having their beginnings also partly on thearc BA k ( u, l ). In fact a general behaviour of a skeleton in such chaotic cases should not differessentially by its chaotic complexity from a chaotic trajectory reminding however rather agigantic road-knot with infinitely many viaducts spanning the billiards boundary on whichthe billiards ball moves. It is obvious that if they exist their identification seems to be notan easy task.Nevertheless the rule (49) can appear to be useful also even in such cases. This is becausea ray beginning with a bundle B k ( u, l ) can come back to it even arbitrarily close to its initialstarting point on A k ( u, l ) (according to the Poincare theorem) not repeating its way. Butthis is enough for writing the ”last quantization condition” (49) where the sum goes now overall bundles of the skeleton passed by the ray.In the next sections we shall focus on the finite number cases of bundles in skeletons notavoiding however billiards with chaotic motions such as the Bunimovich one. We shall discuss16igure 2: The two bundle B (0 , π ; α ) and B A (0 , π ; π − α ) , < α, of the skeleton B in thecircular billiardsalso in sec.8 a possibility of explaining by the skeleton idea phenomena of scars. These arejust phenomena which cannot certainly be described by skeletons with finite structures. In this section we would like to apply the procedure of constructing the SWF’s on theclassical trajectories according to the last section to the circular billiards Fig.2 and to quantizewith these rules the energy of the circular billiard ball.First we have to construct a skeleton B . It follows from the considerations of App.B thatany of it contains only a single bundle B (0 , π ; α ) defined on the full circle L (0 , π ) rays ofwhich make all the same angle α, < α ≤ π , with the tangents to the circle. Its partner B A contains also a single bundle B (0 , π ; π − α ) the rays of which make the angle π − α withthe tangents to the circle. Therefore in the case α = π both the bundles coincide.Motions along such rays conserve of course the angular momentum p cos α so that boththe bundles respect the cylindrical symmetry of the billiards.For these two bundles the corresponding SWF’s are the following (see Fig.2):Ψ ± ( d, s, λ ) = ( d − sin α ) − e ± iλp ( d ± s cos α ) χ ± ( d, s, λ )Ψ ± A ( d, s, λ ) = ( d − sin α ) − e ± iλp ( d ∓ s cos α ) χ ± A ( d, s, λ ) (50)The above SWF’s are defined in the ring 0 ≤ d < sin α, ≤ s ≤ π of the circularbilliards. 17ince both the bundles are defined in the ring mentioned the SWF’s have to satisfy thefollowing uniqueness conditions:Ψ ± ( d, s, λ ) = Ψ ± ( d, s + 2 π, λ )Ψ ± A ( d, s, λ ) = Ψ ± A ( d, s + 2 π, λ ) (51)which lead to λp cos α = − λp cos( π − α ) = m, m = 0 , ± , ± , ... (52)and means the angular momentum quantization together with χ ± ( d, s + 2 π, λ ) = χ ± ( d, s, λ ) χ ± A ( d, s + 2 π, λ ) = χ ± A ( d, s, λ ) (53)We can demand from the SWF’s Ψ ± ( d, s, λ ) and Ψ ± A ( d, s, λ ) to be eigenfunctions Ψ ± m ( d, s, λ )(= Ψ ± ( d, s, λ )) and Ψ A ; ± m ( d, s, λ ) (= Ψ ∓ A ( d, s, λ )) of the respective angular momenta ± m ¯ h , m = 0 , , , ... , where m = λp cos α . Then it follows easily from the forms of the SWF’sconsidered that for the corresponding χ ± m ( d, s, λ ) and χ A ; ± m ( d, s, λ ) we have to have: ∂χ ± m ( d, s, λ ) ∂s = ∂χ A ; ± m ( d, s, λ ) ∂s = 0 (54)i.e. both χ ± m ( d, s, λ ) and χ A ; ± m ( d, s, λ ) are independent of s and therefore they are constanton the circle boundary, i.e. for d = 0.Putting therefore χ ± m (0 , s, λ ) ≡ χ ± m (0 , s, λ ) = − χ A ; ± m (0 , s, λ ) ≡ as ± m ( x, y, λ ) vanishing on the circle boundary and being theangular momentum eigenfunction:Ψ as ± m ( x, y, λ ) = Ψ ± m ( d, s , λ ) + Ψ A ; ± m ( d, s , λ ) m = 0 , , , ... ( x, y ) = (cos s , sin s ) + d ( − sin( α + s ) , cos( α + s )) =(cos s , sin s ) − d ( − sin( − α + s ) , cos( − α + s )) (56)Making further the identificationΨ A ; ± m ( d, s, λ ) = Ψ cont ± m (2 sin α − d, s − α, λ ) , sin α > d ≥ cont ± m ( d, s, λ ) denote Ψ ± m ( d, s, λ ) continued by the caustic r = cos α we get forΨ as ± m ( x, y, λ ) ≡ Ψ as ± m ( r cos Φ , r sin Φ , λ ) (see Fig.2):Ψ as ± m ( r cos Φ , r sin Φ , λ ) = Ψ ± m ( d, s, λ ) + Ψ A ; ± m ( d, − s, λ ) =Ψ ± m ( d, s, λ ) + Ψ cont ± m (2 sin α − d, − s − α, λ ) =(sin α − d ) − (cid:16) − ie ± iλpd ± ims χ ± m ( d, s, λ ) ± e ∓ iλp ( d − α ) ± im (2Φ − s − α ) χ cont ± m (2 sin α − d, − s − α, λ ) (cid:17) = m = 0 , , , ...d = sin α − p r − cos α sin(Φ − s ) = sin αr (sin α − p r − cos α )sin α > d ≥ χ A ; ± m ( d, s, λ ) = e ± i ( λp sin α − mα − π ) χ cont ± m (2 sin α − d, s − α, λ )sin α > d ≥ e ± i ( λp sin α − mα + π ) χ cont ± m (2 sin α, s − α, λ ) = χ ± m (0 , s, λ ) ≡ Z α J △ (cid:16) J − χ ± m ; k ( a, s ) (cid:17) + 2 k X l =0 E k − l +1 χ ± m ; l ( a, s ) ! da = 0 k = 0 , , ... (61)with χ ± m ;0 ( d, s ) ≡ χ ± m ; k +1 ( d, s ) = ± i p Z d J △ (cid:16) J − χ ± m ; k ( a, s ) (cid:17) + 2 k X l =0 E k − l +1 χ ± m ; l ( a, s ) ! dak = 0 , , , ... (62)The integrations in the formulae (61)-(62) go above the singular point d = sin α in the d -plane for the plus sign and below this point for the minus one.A simple conclusion from the quantization conditions (61)-(62) is that:( χ + m ; k ( d, s )) ∗ = χ − m ; k ( d, s ) k = 0 , , , ... (63)so that (cid:0) Ψ as + m ( x, y, λ ) (cid:1) ∗ = Ψ as − m ( x, y, λ ) (64)i.e. the semiclassical energies E in the circular billiards are degenerate with respect to thesign of the angular momentum.Of course the above conclusions are well known as exact for the circular billiards.Therefore in the equations (61)-(62) we can choose only the plus sign in the respectivecalculations to get: λ s E mr − m λ − m arccos mλ √ E = rπ − π α mr = mλ p E mr m = 0 , , , ..., r = 1 , , ... (65)for the ”classical” energy E and the angle α .19t follows from (65) that E r = (cid:16) r − (cid:17) π λ cos α r = 0 E mr = m λ F ( r − m π )cos α mr = F ( r − m π ) m = 1 , , ..., r = 1 , , ... (66)where F − ( x ) = ( √ − x − x arccos x ) x − , < x ≤ D ( t, s ) = J · △ · J − in (68) is bilinear in ∂∂t and ∂∂s with coefficients completely independentof s and it acts in (68) also on the s -independent quantities. In such cases its action is reducedto (for t → x = pt ): h J · △ · J − i red ( x, ∂∂x ) = cos αJ + 1 ! ∂ ∂x + 2 cos αJ ∂∂x + 54 cos αJ −
14 1 J (67)where J = sin α − x .Taking this into account in the second of the eqs. (61) and putting there k = 0 we get: E mr = −
14 sin α mr Z α mr (cid:16) J · △ · J − (cid:17) red · dx = 18 sin α mr (cid:18)
56 cot α mr − (cid:19) (68)where the integration in the x -plane went over upper half-circle with the center at x = sin α mr in the clockwise direction.Similarly for χ mr ;1 ( d ) we get: χ mr ;1 ( d ) = i p Z d
54 cos α mr J mr −
14 1 J mr ! dx + ip E mr d = − i p
512 cos α mr J mr −
512 cos α mr sin α mr −
14 1 J mr + 14 1sin α mr ! + ip E mr d (69)where J mr = sin α mr − d .The higher order terms of the semiclassical expansions for E mr ( λ ) and χ mr ( d, λ ) can beobtain analogously using the recurrent equations (61)-(62).It is to be noticed that the above calculations form a new algorithm for the semiclassicalapproximation method. Consider now the rectangular billiards shown in Fig.3. This billiards is the canonicalexample of the energy quantization problem because of its easiness to be solved by the variable20igure 3: The rectangular billiards with eight ray bundlesseparation method. According to Fig.3 the well known solution to the problem is given bythe following two equations: sin( λp x a ) = 0sin( λp y b ) = 0 (70)with p x + p y = p = 2 E as the result of the following form of the energy eigenfunction:Ψ( x, y ) = C sin( λp x x ) sin( λp y y ) = − C (cid:16) e λp x x + λp y y + e − λp x x − λp y y − e λp x x − λp y y − e − λp x x + λp y y (cid:17) (71)Of course one can always put p x = p cos α, p y = p sin α where α, < α < π , is the anglethe momentum p is inclined to the x -axis.Let us note that the cases α = 0 , π are excluded by the solutions (71). < α < π Let us consider this rectangular billiard example by the method described in the previoussections. Since the absolute values of the momentum components p x , p y are the integrals ofthe classical motion inside the billiards respecting elastic law of bouncing then all bundleswhich should be taken into account are defined by a single angle α, < α < π , which aremade by the rays of the bundle B = B (0 , a ; α ) with the x -axis, see Fig.3.Choosing the case of the angle α shown in Fig.3 the remaining seven bundles of theskeleton B shown in Fig.3 are B = B (0 , a ; π − α ) , B = B ( a, b ; π − α ) , B = B ( a, b ; π +21 ) , B = B ( a + b, a ; π + α ) , B = B ( a + b, a ; 2 π − α ) , B = B (2 a + b, b ; 2 π − α ) , B = B (2 a + b, b ; α ), i.e. the parameter s introduced in sec.3 is counted anticlockwise startingfrom the point (0 ,
0) of Fig.3 (and having negative value if measured clockwise). The bundles B k − , B k , k = 1 , ..., , are defined on the respective sides L k , k = 1 , ..., , of the billiards,i.e. on L = L (0 , a ) , L = L ( a, b ) , L = L ( a + b, a ) , L = L (2 a + b, b ).The skeleton B T coincides exactly with B in the case of the rectangular billiards.Let us note that a number of bundles in the skeletons is obviously independent of a choiceof α , i.e. it is always equal to eight.Note also that unlike the circular billiard bundles the rectangular bundles do not havetheir caustics inside the billiards, but rather in the infinities.The semiclassical wave function Ψ as ( x, y ) constructed according the rules discussed earliershould be the sum at most of the eight SWF’s Ψ + i ( d, s, λ ) , i = 1 , ...,
8, which are constructedon the respective bundles B i , i = 1 , ...,
8, of the skeleton B . But as it follows from Fig.4 onlyfour of them interfere in each point of the rectangular billiards, i.e. we have:Ψ as ( x, y ) = X i =1 Ψ + i ( d i , s i , λ ) = X i =1 Ψ + k i ( d i , s i , λ ) x = x ( s i ) + d i cos α ( s i ) y = y ( s i ) + d i sin α ( s i ) i = 1 , ..., α ( s i ) are the angles the rays in Fig.4 make with the x -axis and ( x ( s i ) , y ( s i )) are thepoints of the rectangular boundary from which the rays start. The parameters s i as previouslymeasure distances of the boundary points ( x ( s i ) , y ( s i )) from the point (0 ,
0) in anticlockwisedirection along the boundary. The corresponding Jamaicans are J ( t, s i ) = p − sin α ( s i ), i.e.are constant but discontinues. Therefore they will be included into the χ -factors containedin the SWF’s.Let us note that Ψ as ( x, y ) can be discontinue each time the point ( x, y ) crosses the thinblack lines in Fig.3. being the boundary of different bundles. Let us enumerate these linesby l i , i = 1 , ..., , starting from the line emerging from the point (0 ,
0) and continuinganticlockwise.It is then easy to see that to ensure the continuity of Ψ as ( x, y ) in each point in thebilliards it is enough to identify the respective χ -coefficients on the respective lines definingthem elsewhere in the billiards as continuous functions of s and d . We get such a continuityputting: χ ± ( d, , λ ) = χ ± ( d, , λ ) χ ± ( d, a, λ ) = χ ± ( d, a, λ ) χ ± ( d, a + b, λ ) = χ ± ( d, a + b, λ ) χ ± ( d, a + b, λ ) = χ ± ( d, a + b, λ ) (73)Of course Ψ as ( x, y ) vanishes on the rectangular boundary, i.e. we have to have:Ψ as (0 , y ) = Ψ as ( a, y ) = Ψ as ( x,
0) = Ψ as ( x, b ) = 00 < x < a, < y < b (74)22igure 4: Four solutions Ψ + i ( d i , s i ) , i = 1 , ...,
4, meeting at the point ( x, y )The conditions (74) expressed in terms of the SWF’s Ψ ± i ( d, s, λ ) , i = 1 , ..., , take thefollowing detailed forms: Ψ +2 k − (0 , s, λ ) + Ψ − k (0 , s, λ ) = 0Ψ +2 k (0 , s, λ ) + Ψ − k − (0 , s, λ ) = 0 s ∈ L k , k = 1 , ..., − i (0 , s, λ ) , i = 1 , ...,
8, with the respective Ψ + j ( d j ( s ) , f j ( s ) , λ ) , j =1 , ...,
8, it should be noticed that for each considered trajectory family there is no causticinside the rectangular billiards so that each integral R K p x dx + p y dy along the close loop K lying inside the billiards or on its boundary has to vanish. As a consequence of thisthe identifications we are talking about are then reduced just to the identifications of thecorresponding χ -factors, i.e the possible phase factors are factored out. Therefore we getsuccessively for the conditions (75) to be satisfied:1. s > , r ( s ) ∈ I, − s tan α ∈ XII, a + b − b cot α − r ( s ) ∈ V IIχ − (0 , s, λ ) = χ +4 (cid:18) b sin α , a + b − b cot α − s, λ (cid:19) χ − (0 , s, λ ) = χ +7 (cid:18) s cos α , − s tan α, λ (cid:19) (76)and hence and from (75) χ (0 , s, λ ) + χ (cid:18) s cos α , − s tan α, λ (cid:19) = 023 (0 , s, λ ) + χ (cid:18) b sin α , a + b − b cot α − s, λ (cid:19) = 0 (77)In the last two equations the plus signs at χ + j , j = 1 , ...,
8, have been dropped and thisconvention will be kept in further equations if not leading to misunderstandings.2. s > , r ( s ) ∈ II, − s tan α ∈ X + XI, a + ( a − s ) tan α ∈ V + V I ) χ − (0 , s, λ ) = χ +4 (cid:18) a − s cos α , a + ( a − s ) tan α, λ (cid:19) χ − (0 , s, λ ) = χ +7 (cid:18) s cos α , − s tan α, λ (cid:19) (78)and hence and from (75) χ (0 , s, λ ) + χ (cid:18) s cos α , − s tan α, λ (cid:19) = 0 χ (0 , s, λ ) + χ (cid:18) a − s cos α , a + ( a − s ) tan α, λ (cid:19) = 0 (79)3. r ( s ) ∈ III, s > , ( − b − s + b cot α ∈ IX, a + ( a − s ) tan α ∈ IVχ − (0 , s, λ ) = χ +4 (cid:18) a − s cos α , a + ( a − s ) tan α, λ (cid:19) χ − (0 , s, λ ) = χ +6 (cid:18) b sin α , − b − s + b cot α, λ (cid:19) (80)and hence χ (0 , s, λ ) + χ (cid:18) b sin α , − b − s + b cot α, λ (cid:19) = 0 χ (0 , s, λ ) + χ (cid:18) a − s cos α , a + ( a − s ) tan α, λ (cid:19) = 0 (81)4. s > , r ( s ) ∈ IV + V + V I, a − ( s − a ) cot α ∈ II + III,a + b + ( a + b − s ) cot α ∈ V II + V IIIχ − (0 , s, λ ) = χ +6 (cid:18) a + b − s sin α , a + b + ( a + b − s ) cot α, λ (cid:19) χ − (0 , s, λ ) = χ +1 (cid:18) s − a sin α , a − ( s − a ) cot α, λ (cid:19) (82)and hence χ (0 , s ) e − iλpa cos α + χ (cid:18) s − a sin α , a − ( s − a ) cot α, λ (cid:19) e iλpa cos α = 0 χ (0 , s, λ ) e − iλpa cos α + χ (cid:18) a + b − s sin α , a + b + ( a + b − s ) cot α, λ (cid:19) e iλpa cos α = 0 (83)24. s > , r ( s ) ∈ V II, a + b − b cot α − r ( s ) ∈ I, a + b − ( s − a − b ) tan α ∈ V Iχ − (0 , s, λ ) = χ +1 (cid:18) b sin α , a + b − b cot α − s, λ (cid:19) χ − (0 , s, λ ) = χ +3 (cid:18) s − a − b cos α , a + b − ( s − a − b ) tan α, λ (cid:19) (84)and hence χ (0 , s, λ ) e − iλpb sin α + χ (cid:18) b sin α , a + b − b cot α − s, λ (cid:19) e iλpb sin α = 0 χ (0 , s, λ ) e − iλpb sin α + χ (cid:18) s − a − b cos α , a + b − ( s − a − b ) tan α, λ (cid:19) e iλpb sin α = 0 (85)6. s < , r ( s ) ∈ V III, − b − ( b + s ) tan α ∈ X + XI + XII,a + b − ( a + b + s ) tan α ∈ IV + Vχ − (0 , s ) = χ +1 (cid:18) − b − s cos α , − b − ( b + s ) tan α, λ (cid:19) χ − (0 , s ) = χ +2 (cid:18) a + b + s cos α , a + b − ( a + b + s ) tan α, λ (cid:19) (86)and hence χ (0 , s, λ ) e − iλpb sin α + χ (cid:18) − b − s cos α , − b − ( b + s ) tan α, λ (cid:19) e + iλpb sin α = 0 χ (0 , s, λ ) e − iλpb sin α + χ (cid:18) a + b + s cos α , a + b − ( a + b + s ) tan α, λ (cid:19) e iλpb sin α = 0 (87)7. s < , r ( s ) ∈ IX, − b − ( b + s ) tan α ∈ X, b cot α − b − r ( s ) ∈ IIIχ − (0 , s ) = χ +8 (cid:18) − b − s cos α , − b − ( b + s ) tan α, λ (cid:19) χ − (0 , s ) = χ +2 (cid:18) b sin α , b cot α − b − s, λ (cid:19) (88)and hence χ (0 , s, λ ) e − iλpb sin α + χ (cid:18) − b − s cos α , − b − ( b + s ) tan α, λ (cid:19) e + iλpb sin α = 0 χ (0 , s, λ ) e − iλpb sin α + χ (cid:18) b sin α , b cot α − b − s, λ (cid:19) e iλpb sin α = 0 (89)8. s < , r ( s ) ∈ X + XI + XII, − s cot α ∈ I + II, − b − ( s + b ) cot α ∈ V III + IXχ − (0 , s, λ ) = χ +5 (cid:18) b + s sin α , − b − ( s + b ) cot α, λ (cid:19) χ − (0 , s, λ ) = χ +2 (cid:18) − s sin α , − s cot α, λ (cid:19) (90)25igure 5: The solution Ψ +8 ( t, s ) being carried by the successive bundles B → B → B → B → B → B → B and hence χ (0 , s, λ ) + χ (cid:18) − s sin α , − s cot α, λ (cid:19) = 0 χ (0 , s, λ ) + χ (cid:18) b + s sin α , − b − ( s + b ) cot α, λ (cid:19) = 0 (91)Solving the last set of the quantization conditions we should get the solutions for theenergy levels and for the corresponding Ψ as ( x, y ).However the above conditions do not provide us with the solutions for χ i (0 , s, λ ) , i =1 , ...,
8, in some algebraic forms. Rather they show how a solution defined on some bundlepropagates along its rays to be transformed by bouncing off the rectangular boundary intoanother bundle and finally after a finite number of such bouncings achieving its mother bundleand repeating this process infinitely.To get a knowledge what really happens by such bouncings consider for example thegreen color ray along which the solution Ψ +8 ( t, s ) propagates, shown in Fig.5 emerging fromthe point s, r ( s ) ∈ X , of the rectangular boundary. This particular ray bounces into theblue-light one and next again into the green one. Taking into account first the conditions(85) and next (81) we get: χ (0 , ( s + 2 b ) cot α, λ ) = − χ (cid:18) b sin α , − b − ( s + b ) cot α, λ (cid:19) = χ cont (cid:18) b + s sin α , s, λ (cid:19) e +2 iλpb sin α , r ( s ) ∈ X (92)26nd continuing the bounces to achieve the green ray again we get: χ (0 , ( s + 4 b ) cot α − a, λ ) = χ cont ( D ( s ) , s, λ ) e iλpb sin α +2 iλpa cos α , r ( s ) ∈ X (93)where D ( s ) is the total distance the green ray has passed from the point s to the point( s + 4 b ) cot α − a bouncing multiply from the boundary.But taking into account the first of the identification (73) and the identification (49) wehave to have: χ (0 , ( s + 2 b ) cot α, λ ) = χ cont (cid:18) b + s sin α , s, λ (cid:19) χ (0 , ( s + 4 b ) cot α − a, λ ) = χ cont ( D ( s ) , s, λ ) r ( s ) ∈ X (94)so that e +2 iλpb sin α = 1 e iλpb sin α +2 iλpa cos α = 1 (95)and consequently: λpb sin α = mπλpa cos α = nπm, n = 1 , , ... (96)Taking the limit λ → ∞ in (94) we get: χ , (( s + 2 b ) cot α ) = χ , (( s + 4 b ) cot α − a ) = χ , ( s ) (97)If tan α = mn ab is not rational none of the arrival points can be repeated, i.e. the corre-sponding trajectory is not closed (periodic), and the (infinite) set of such points is dense onthe side L of the billiards. Therefore the equations (97) show that χ , ( s ) has to be not onlya constant of motion but also s -independent since this equation can be written in infinitelymany points of L densely distributed on it.The last conclusion is valid also of course for the remaining χ k, ( s ) , k = 1 , ..., , and as itcan be easily concluded from (14) and App.D all the coefficients χ k ( d, s, λ ) , k = 1 , ..., , arethen also constant. Putting therefore χ ( d, s, λ ) ≡ χ k ( d, s, λ ) ≡ , k = 1 , , , χ k ( d, s, λ ) ≡ − , k = 2 , , , x, y ) of Fig.4 for the SWF (72) we get:Ψ as ( x, y ) = Ψ +2 ( d , s ) + Ψ +5 ( d , s ) + Ψ +7 ( d , s ) + Ψ +8 ( d , s ) = X l =1 ( − l e iλp t l + iλpf l ( s l ) (99)where t l and f l ( s l ) , l = 1 , ..., x = x ( s l ) + pt l cos α ( s l ) y = y ( s l ) + pt l sin α ( s l ) α ( s l ) = α, π − α (100)27owever making use of the independence of the phase integral R ( x,y )(0 , p x dx + p y dy of theintegration paths we get for the particular terms in the sum in (99): e iλp t + iλpf ( s ) = e iλ ( − p x x + p y y ) e iλp t + iλpf ( s ) = e iλ ( − p x x − p y y ) e iλp t + iλpf ( s ) = e iλ ( p x x − p y y ) e iλp t + iλpf ( s ) = e iλ ( p x x + p y y ) (101)and finally:Ψ as ( x, y ) = − e iλ ( − p x x + p y y ) + e iλ ( − p x x − p y y ) − e iλ ( p x x − p y y ) + e iλ ( p x x + p y y ) = − p x x ) sin( p y y ) (102)reproducing in this way the exact result (71).The cases when tan α = mn ab is rational are possible only if ab is rational. For maintainingthe results for the irrational ab one can argue relying on a continuity of all the investigatedquantities considered as functions of a, b , since each point ( a, b ) with rational value of ab isdensely surrounded by the ones for which ab is irrational. α = 0 , π - the bouncing mode skeletons It is surprising that the cases α = 0 , π are not allowed by the representation (71) of theSWF leading to the totally vanishing solutions while there are still the bundle skeletons withthese angles on which non vanishing identically SWF’s can be constructed. Therefore weshould get some new knowledge about the semiclasscical method developed here consideringthese cases known as the bouncing modes.Consider the case α = π . The second case will be analogous.There are two bundles in this case which the skeleton B is consisting of. The one B withits rays directed up and starting from the side L and the second B with rays directed downstarting from the side L (Fig.6). The skeleton B A is identical with B .For these particular cases of bundles rays for both the bundles will be positioned by thesame parameter s measuring a distance of a ray from the y -axis along the corresponding sides L and L . Therefore for the corresponding SWF’s we get:Ψ ± ( d, s, λ ) = e ± iλpd χ ± ( d, s, λ )Ψ ± ( b − d, s, λ ) = e ± iλp ( b − d ) χ ± ( b − d, s, λ )0 ≤ d ≤ b, < s < a (103)For the SWF Ψ as ( x, y, λ ) we have:Ψ as ( x, y, λ ) = Ψ +1 ( d, s, λ ) + Ψ +3 ( b − d, s, λ ) = Ψ − ( b − d, s, λ ) + Ψ − ( d, s, λ )( x, y ) = ( s, d ) (104)together with the following identifications:Ψ ± ( b − d, s, λ ) = Ψ ∓ ( d, s, λ )Ψ +1 (0 , s, λ ) = − Ψ − (0 , s, λ )Ψ +3 (0 , s, λ ) = − Ψ − (0 , s, λ ) (105)28igure 6: The two bouncing mode bundles B and B of the vertical skeleton in the rectan-gular billiardsso that: χ ± ( b − d, s, λ ) = e ∓ iλpb χ ∓ ( d, s, λ ) χ +1 (0 , s, λ ) = − χ − (0 , s, λ ) χ +3 (0 , s, λ ) = − χ − (0 , s, λ ) (106)It follows from the last equations that: χ +3 (0 , s, λ ) = − χ − (0 , s, λ ) = − e iλpb χ +1 ( b, s, λ ) (107)so that: χ +3 ( b, s, λ ) = − e iλpb χ +1 (2 b, s, λ ) (108)and further: χ +1 (0 , s, λ ) = − χ − (0 , s, λ ) = − e iλpb χ +3 ( b, s, λ ) = e iλpb χ +1 (2 b, s, λ ) = e iλpb χ +1 (0 , s, λ ) (109)since χ +1 (2 b, s, λ ) = χ + ,cont (0 , s, λ ) = χ +1 (0 , s, λ ).Therefore: e iλpb = 1 (110)and Ψ as ( x, y, λ ) = e iλpd χ +1 ( d, s, λ ) + e − iλpd e iλpb χ +3 ( b − d, s, λ ) = e iλpd χ +1 ( d, s, λ ) − e − iλpd χ +1 (2 b − d, s, λ ) (111)29e have of course Ψ as ( x, , λ ) = Ψ as ( x, b, λ ) = 0 by construction. But we have to havealso Ψ as (0 , y, λ ) = Ψ as ( a, y, λ ) = 0, i.e. we have to have: e iλpd χ +1 ( d, , λ ) − e − iλpd χ +1 (2 b − d, , λ ) = 0 e iλpd χ +1 ( d, a, λ ) − e − iλpd χ +1 (2 b − d, a, λ ) = 0 (112)Therefore from (112) in the limit λ → ∞ we have: e iλpd χ +1 , (0) − e − iλpd χ +1 , (0) = 0 e iλpd χ +1 , ( a ) − e − iλpd χ +1 , ( a ) = 0 (113)so that χ +1 , (0) = χ +1 , ( a ) = 0 (114)Next let us invoke the second of the equations (14) to get in the considered case for d = 2 b : χ +1 , (2 b, s ) = χ +1 , (0 , s ) + i bp d χ +1 , ( s ) ds + 2 E χ +1 , ( s ) ! (115)so that d χ +1 , ( s ) ds + 2 E χ +1 , ( s ) = 0 (116)since χ +1 , (2 b, s ) = χ +1 , (0 , s ).The obvious solution of the last equation satisfying the boundary conditions (114) is: χ +1 , ( s ) = A sin( p E s ) p E a = mπ, m = 1 , , ... (117)Coming back to the second of the equations (14) we can conclude that χ +1 , ( d, s ) again isindependent of d . Passing next to the third of the equations (14) and repeating argumentssimilar to those which led us to (115)-(116) we get the following explicit dependence of χ +1 , ( s )on s : χ +1 , ( s ) = A sin( p E s ) + B cos( p E s ) + E A s √ E cos( p E s ) (118)The boundary conditions χ +1 , (0) = χ +1 , ( a ) = 0 enforce however B = E = 0.Using again (14) and the inductive arguments we come to the conclusion that χ +1 ( d, s, λ )is d -independent and coefficients of its semiclassical series have the form: χ +1 ,k ( s ) = A k sin( p E s ) , k = 0 , , ... (119)so is the form of χ +1 ( s, λ ) itself, i.e. χ +1 ( s, λ ) = A ( λ ) sin( p E s ) (120)Therefore coming back to (111) we get:Ψ as ( x, y, λ ) = ( e iλpd − e − iλpd ) χ +1 ( s, λ ) = 2 iA ( λ ) sin( λpd ) sin( p E s ) (121)30igure 7: An ”arbitrary” broken rectangular billiardswhich again is the result of the previous way of the rectangular billiard energy quantization.But now the energy E is given by the (finite) semiclassical series: E = 12 p + E λ = 12 (cid:18) nπλb (cid:19) + (cid:18) mπλa (cid:19) ! , m, n = 1 , , ... (122)Therefore we get a surprising result that SWF’s in the rectangular billiards can be builtequivalently by the following two ways:1. on the skeletons which rays are inclined to the billiard sides with corresponding anglesdefined by the quantization conditions (95) - in this case the χ -coefficients of the SWF’sare simply constant on the rectangular boundary;2. on one of the two skeletons which rays are perpendicular to one of the rectangular sides- in this case the corresponding χ -coefficients vary along the sides perpendicular to therays and vanish on the sides parallel to them.But a more important conclusion which follows from the results got in this section isthat a relation between the form of skeletons and the SWF’s and the corresponding energyspectrum which seemed to be suggested by the first part of the section is rather illusory sincea full description of these quantities can be also obtained using a single skeleton only at leastin the case of the rectangular billiards. By a broken rectangular billiards we understand the one on Fig.7, i.e. with some numberof rectangular bays and peninsulas. 31igure 8: A single bay rectangular billiardsTo illustrate the way of energy quantization in such billiards we shall consider first theone with a single peninsula shown in Fig.8. The corresponding procedures are analogous tothe previous ones. One needs to construct additional four bundles relating with the peninsulasides.Although both the previous methods of the sections 5.1 and 5.2 are equivalent leading usto the same results the method of sec.5.2 seems however to be simpler and more instructivein applications to more complicated cases of billiards.We assume from the very beginning that all the sides of the billiards from Fig.10, i.e. a = 1 , b, a ′ , b ′ are commensurate. This assumption can always be satisfied even if the sidesare expressed by irrational number just by respective approximations of the latter by rationalones with arbitrary accuracies. We proceed as follows.We construct two vertical skeletons. One for the rectangle OAEF and the second for
ABCD .For the first skeleton according to (121) we get the following SWF:Ψ as ( x, y, λ ) = A sin( λpy ) sin( p E x )0 ≤ x < a ′ , ≤ y ≤ bλpb = mπ, m = 1 , , ... p E a ′ = nπ, n = 1 , , ... (123)while for the second:Ψ as ( x, y, λ ) = A sin( λp ′ y ′ ) sin (cid:18)q E ′ (1 − x ′ ) (cid:19) a ′ < x ′ ≤ , ≤ y ′ ≤ b ′ λp ′ b ′ = kπ, k = 1 , , ... (124)32o get Ψ as ( x, y, λ ) for the total rectangular we have to match both the previous ones onthe segment AD of Fig.8. Since this matching has to be valid for 0 ≤ y ′ ≤ b ′ then as it followseasily we have to have p = p ′ so that E = E ′ because of (122), i.e. the vertical wave lengthsof both the matched solutions have to be the same. Therefore we get: A sin( p E a ′ ) = A sin (cid:16)p E (1 − a ′ ) (cid:17) A cos( p E a ′ ) = − A cos (cid:16)p E (1 − a ′ ) (cid:17) (125)so that √ E = lπ, l = 1 , , ..., and A = ( − l +1 A .Therefore the procedure leads us to the following quantization conditions for the energy E nm : 1 n = n Λ x , bm = m Λ y x = 2 π √ E , Λ y = 2 πλpE nm = 12 p + 2 Eλ = 2 π λ x + 1Λ y ! = π λ n n + m m b ! m, n = 1 , , ... (126)where Λ x , Λ y are the wave lengths of rays in the horizontal and vertical skeletons respectivelyand n , m are the smallest integers satisfying l = n a ′ and k b = m b ′ where l , k are alsointegers.The respective SWF’s are the following:Ψ asnm ( x, y, λ ) = ( A sin πx Λ x sin πy Λ y = A sin( nn πx ) sin (cid:0) mm b y (cid:1) ( x, y ) ∈ D br x, y ) / ∈ D br (127)where D br denotes the domain of the x, y -plane occupied by the broken rectangular of Fig.8.One can easily realize that the last results can be easily generalized to any broken rectan-gular billiards. A little bit surprising is that the formulae (126) for the energy and (127) forthe wave functions remain unchanged for any such a billiards while a number of conditionsthe wave lengths Λ x and Λ y have to satisfy filling the vertical and horizontal skeletons byinteger numbers of their halves is increasing respectively to numbers of bays and peninsulasforming the sides od such billiards. The semiclassical quantization of the rectangular billiards in sec.5.2 shows that it is possi-ble to quantize semiclassically in the similar way some specific configurations of the SWF’s inmore complicated billiards also classically chaotic such as polygons or Bunimovich billiards.Such easy opportunities appear if billiards to be considered possess boundaries which allow usfor easy constructions of corresponding skeletons. Some simple examples of such billiards areprovided by a parallelogram, a trapezium, a pentagon shown in Fig.9 or by the Bunimovichstadium - Fig.10, and its simple generalizations - Fig.11.33igure 9: A parallelogram, a trapezium and a pentagon billiards with the bouncing modeskeletonsFigure 10: The Bunimovich billiards with the bouncing mode skeleton and with the scargenerating bundles associated with the horizontal periodic orbit34igure 11: A simple generalization of the Bunimovich billiards with the corresponding bounc-ing mode skeletons 35he corresponding bundle skeletons are shown on the respective figures.The constructions of the SWF’s corresponding to the skeletons shown on the figurescoincide with those for the rectangular billiards and therefore these SWF’s are the following:Ψ as, + ( x, y, λ ) = A sin( λp x ) sin( λp y ) ,E = 12 (cid:16) p + p (cid:17) λp a = mπ, λp b = nπ, m, n = 1 , , ... (128)for the skeletons perpendicular to the x -axis and defined in the respective rectangles a × b of the figures while outside the skeletons Ψ as, + ( x, y, λ ) = 0, i.e. those regions are classically”forbidden” and the corresponding wave functions vanish there exponentially if λ → ∞ . .For the Bunimovich-like billiards of Fig.11 it should be clear that bouncing modes can beexcited independently in each of its rectangular part but also simultanuously in both partstunning the modes respectively.Formulae like (128) have been suggested by Burq and Zworski [20] by completely differentbut mathematically rigorous arguments. It was shown in the previous section that the idea of the skeletons seems to be effectivein solving some simple situations of quantum phenomena related semiclassically with thechaotic dynamics. Nevertheless obvious difficulties in effective constructions of skeletons inthe cases of chaotic dynamics seems to limit seriously its applications. Despite this one cantry to understand with it the scar phenomena noticed by Heller [14] and investigated by thelatter author and others [15, 19, 20, 21]. Although in examples discussed below we do notconstruct closed skeletons but yet supposing their existence in these cases allows us at leasthalf-qualitatively to understand the scar phenomena and to make some predictions for theirexistence in some cases of billiards and their absence in others.Consider for example the Bunimovich stadium billiards of Fig.10 and the isolated unstablehorizontal periodic orbit linking the top points A and B of the stadium. It is known [14] thatthe there is a mode for which its wave function takes significantly larger values around theorbit than far away of it, i.e. the orbit signals its existence by such a ”scar”. This andsimilar scar phenomena have not got their full description although there were many effortsand approaches to do it [15, 19, 20, 21].Of course it is not easy not only to construct a skeleton carrying such a mode but evento prove its existence. Nevertheless assuming the latter we can expect that such a skeletonwill be symmetric horizontally and vertically as well as its bundles will contain the horizontalperiodic orbit.Consider one of such bundles emerging from a vicinity of the top point A on Fig.10. Evenif its rays are divergent with respect to the periodic orbit the rays which are very close to itare also almost parallel to it, i.e. after the reflection by the opposite semicircle of the billiardsthey are transformed into convergent bundle focused close to the focal point of the reflectingsemicircle. It means that central parts of almost all such bundles of the skeleton containingthe horizontal periodic orbit have to be convergent and have to pass close to focal points ofthe semicircles.Assume the radii of the semicircles to be equal to 1 while according to Fig.10 the flatparts of the stadium have the length a each. It is clear that each such a convergent bundle36ill generate after the reflection a new bundle the central part of which is again convergent.On this new bundle a new χ -factor defined on it will however be weakened according to (40)by the factor (cid:12)(cid:12)(cid:12) ∂h ( s ; u,l ) ∂s (cid:12)(cid:12)(cid:12) − ≈ (2 a + 3) − . Therefore if the starting bundle has the factor χ as its zeroth order semiclassical approximation then after n subsequent reflections it will beweakened by the factor (2 a + 3) − n . However the central parts of all the reflected bundleswill remain close to the periodic orbit so that SWF’s defined on them will interfere in aninfinite number of them close to the orbit.Easy calculations of the contributions coming from all such bundles lead us to the followingform of the regular scarring part of the semiclassical wave function on the horizontal periodicorbit of Fig.10 in the JWKB approximation: Ψ JW KBscar ( x,
1) = e iλp ( a +2) − q e iλp ( a +2) (cid:16) (2 x + 1) − e − iλp ( a − x +1) − (2( a − x ) + 1) − qe + iλp ( a − x +1) (cid:17) χ (1) − ≤ x ≤ a + 1 , x = − , a + 12 , q = (2 a + 3) − (129)where arg(2 x + 1) = π for x < − and the point x = − is avoided clockwise and above itwhen x moves to the values larger than − and the point A has been chosen as the initialone for the s -variable so that s ≈ y − JW KBscar ( x,
1) does not vanish for x = − , a + 1, see our comment below). There are still infinitely many bundles contributingto the SWF coming from the parts of the bundles discussed above scattered away of theircentral parts. However it seems reasonable to assume that just such bundles scattered uponthe whole billiards interfere everywhere chaotically, i.e. typically as in the cases without scarsand satisfying Berry’s conjecture [26]. Some order in such interfering can be organized as wecan try to argue by the presence of periodic orbits supported by the focusing properties ofthe billiards boundary in vicinities of top points, i.e. the points where periodic orbits touchthe billiards boundary.The last condition, i.e. the focusing properties of the billiards boundary at the top pointsseems to be essential for the scar phenomenon to appear. One can be convinced of itsnecessity by considering an ”anti-Bunimovich” stadium, i.e. the stadium which semicircularparts instead of being concave are convex for the billiards, see Fig.12. The horizontal periodicorbit still exists but a skeleton with the properties discussed earlier certainly does not, i.e.the bundles containing this periodic orbit can be only scattered so that the only rays of thesebundles which come back close to the periodic orbit is the orbit itself. All the remaining rayseven close to the periodic orbits are scattered away passing by the virtual focus points of theconvex semicircles.The results of Barnett’s calculations demonstrated by Sarnak [24, 25] seem also to confirmthis conclusion.Let us note however that the above analysis is not sufficient to estimate the energy cor-responding to the mode described above even its lowest JWKB approximation investigatingonly a vicinity of the horizontal periodic orbit . This is because after every reflection thecorresponding half of the horizontal periodic orbit belongs to a new bundle, i.e. this orbitis in fact not closed on the skeleton considered and moreover it does not end in the initialbundle (this is why it is not periodic inside the skeleton). Because of that this classicallyperiodic orbit cannot be used to write on it ”the last quantization condition” of the form37igure 12: The anti-Bunimovich billiards with the bouncing mode skeleton deprived of thescar generating bundles associated with the horizontal periodic orbit(49). In fact such a single periodic orbit possibility of determining the energy would questionthe role of the eigenfunction boundary conditions in forming its corresponding eigenvalue. We have shown in this paper that it is possible to formulate the semiclassical descriptionof the quantum billiards eigenvalue problems in the semiclassical wave function language.We have used in principle the approach of Maslov and Fedoriuk [6] modifying it howeverby the way of moving through caustics. The corresponding procedure has been described inApp.B. where the idea of continuing semiclassical wave functions by coustic singularities onthe complex t -plane was developed. It was shown there a close relation between a signatureof a SWF and a path along which the SWF has to avoid a caustic singularity on the t -plane.We have also modified the Maslov and Fedoriuk procedure by constructing respectiveLagrange manifolds not as smooth, deprived of boundaries tori-like surfaces but rather in theform of skeletons. The skeleton idea has appeared to be sufficiently flexible to gather quantumsystems at least in principle uniformly independently of the kind of their semiclassiclal limits,i.e. whether these limits are integrable or chaotic. Nevertheless in the last cases constructionsof the corresponding skeletons seem to be rather difficult except these obvious chaotic billiardsboundary configurations which have been considered in this paper.It is worth to stress also at this summary that the forms of skeletons as a set of ray bundlessatisfying the geometrical optics law of the mirror-like reflection from the billiards boundarywas not a matter of a free choice but a necessity of satisfying the boundary conditions by theSWF’s as it was shown in App.A.No less important was establishing that in the semiclassical calculations the zeroth order38igure 13: The example of the four bundles skeleton in the right angle triangular billiards onwhich none semiclassical eigenfunction can be definedterms of SWF’s are classical integrals of motion. This has allowed us to close the correspond-ing calculations.Using known examples of the classically integrable billiards such as the circular andrectangular ones and the broken rectangular billiards as some variants of the rectangularones we have demonstrated the effectiveness of the skeleton method in the quantization ofthe systems mentioned. In particular the semiclassical calculations of energy levels performedin sec.5 have developed a new algorithm of finding these levels to any order in Planck constantfor the case of the circular billiards (i.e. the cylindrical infinite well).Applications of the method to other classically integrable billiards systems also in higherthan two dimensions are in preparations.We have shown also how the skeleton method allows us to describe almost trivially thebouncing ball modes in the Bunimovich billiards and similar ones.We have also discussed a possible explanation of the scar phenomena by the skeleton ideashowing a close relation of these phenomenta to focusing properties of billiards boundary invicinity of the top points of the periodic orbits carrying scars. A lack of focussing propertiesby the billiards boundary in vicinity of the top points should exclude the existence of scarsaround the corresponding periodic orbits.On the other hand it should be stressed that constructing a desired skeleton does notguarantee a construction on it a SWF satisfying desired necessary boundary and other condi-tions. A simple example of such a situation is a rectangular triangel billiards shown on Fig.13.There is a skeleton collected of four bundles as shown in the figure. It is clear however thatnone SWF vanishing on the boundary of the billiards can be built on this skeleton.39 ppendix A In this appendix we are going to show, that the geometrical optics rule of reflections ofrays off the billiards boundary is a consequence of demands of vanishing on the boundary ofthe linear combination (35) accompanied by the conditions (36) and (37). Namely considerthe following superposition of SWF’s:Ψ ask ( x, y ; u, l ; λ ) = Ψ σ k, ( d , s ; u, l ; λ ) + Ψ σ k, ( d , s ; u, l ; λ ) = J − k, ( d , s ; u, l ) e iσ k ( d + R s cos α k, ( s ′ ; u,l ) ds ′ ) χ σ k, ( d , s ; u, l ; λ ) + J − k, ( d , s ; u, l ) e iσ k ( d + R s cos α k, ( s ′ ; u,l ) ds ′ ) χ σ k, ( d , s ; u, l ; λ ) (130)with r ≡ [ x, y ] = r k, ( d , s ; u, l ) = r ( s ) + d ( s ; u, l ) = r k, ( d , s ; u, l ) = r ( s ) + d ( s ; u, l ) r k, ( d, s ; u, l ) ∈ B k ( u, l ) , r k, ( d, s ; u, l ) ∈ B ′ k ( u, l ) , B k ( u, l ) = B ′ k ( u, l ) (131)i.e. the SWF’s Ψ σ k, ( d , s ; u, l ; λ ) and Ψ σ k, ( d , s ; u, l ; λ ) are defined respectively on the bun-dles B k ( u, l ) and B ′ k ( u, l ) with D k ( u, l ) ∩ D ′ k ( u, l ) = ⊘ interfering in the crossing point [ x, y ]of two rays r k, ( d , s ; u, l ) and r k, ( d , s ; u, l ) belonging to the respective bundles.Therefore the condition for Ψ ask ( x, y, λ ) to vanish on A k ( u, l ) is: J − k, (0 , s ) e ikσ R s cos α k, ( s ′ ; u,l ) ds ′ χ σ k, ( s ; u, l ; λ ) + J − k, (0 , s ) e ikσ R s cos α k, ( s ′ ; u,l ) ds ′ χ σ k, ( s ; u, l ; λ ) = 0 r ( s ) ∈ A k ( u, l ) (132)Because of the k -dependence the last relation can be satisfied if and only if: σ Z s cos α k, ( s ′ ; u, l ) ds ′ = σ Z s cos α k, ( s ′ ; u, l ) ds ′ , r ( s ) ∈ A k ( u, l ) (133)It is easy to see however that there are only two solutions of the last condition: α k, ( s ; u, l ) ≡ α k, ( s ; u, l ) f or σ = σ α k, ( s ; u, l ) ≡ π − α k, ( s ; u, l ) f or σ = − σ r ( s ) ∈ A k ( u, l ) (134)The first solutions are however uninteresting identifying the bundles in a given segmentand consequently leading to the solutions vanishing identically on A k ( u, l ).Putting α k, ( s ; u, l ) ≡ α ( s ; u, l ) and σ = − σ we get from the second solution and from(132): χ − σk, ( s ; u, l ; λ ) = − χ σk, ( s ; u, l ; λ ) ≡ χ k ( s ; u, l ; λ ) r ( s ) ∈ A k ( u, l ) (135)40o that the combination (35) becomes:Ψ ask ( x, y ; u, l ; λ ) = Ψ σk, ( d , s ; u, l ; λ ) − Ψ − σk, ( d , s ; u, l ; λ ) = J − k, ( d , s ; u, l ) e iσk ( d + R s cos α ( s ′ ; u,l ) ds ′ ) χ σk, ( d , s ; u, l ; λ ) − J − k, ( d , s ; u, l ) e − iσk ( d + R s cos α ( s ′ ; u,l ) ds ′ ) χ − σk, ( d , s ; u, l ; λ ) r ( s ) ∈ A k ( u, l ) (136)where χ k σ ( d, s ; u, l ; λ ) , σ = ± , are given by (31) with χ σk (0 , s ; u, l ; λ ) ≡ χ k ( s ; u, l ; λ ).The last result shows that Ψ ask ( x, y ; u, l ; λ ) vanishing on A k ( u, l ) has to be representedsemiclassically by a combination of at least two SWF’s of opposite signatures and such thatif Ψ σk, ( d, s ; u, l ; λ ) is defined on the bundle B k ( u, l ) then the second SWF Ψ − σk, ( d, s ; u, l ; λ )has to be defined on the bundle B Ak ( u, l ). Appendix B
In this appendix we explore the well known properties of the circular billiards to establishthe way of avoiding the singular caustic points on the t -plane as well as to confirm the waygiven by (38) by which two SWF’s have been matched.The circular billiards is of course the well known case of the two dimensional infinitely deepcylindrical potential well which energy spectrum is easily obtained by solving the stationarySchr¨odinger equation (SSE) for this case by the variable separation method performed in thecylindrical coordinates. In the latter coordinates the radial part of the SSE is the following:Ψ ′′ ( r ) + 1 r Ψ ′ ( r ) + λ E − m λ r ! Ψ( r ) = 0 (137)where the separation constant m = 0 , ± , ... , is the angular momentum quantum number, λ = ¯ h − and we have put 2 M = 1 where M is the billiard ball mass.After the substitution Ψ( r ) = ψ ( r ) r we get: ψ ′′ ( r ) + λ E − m − λ r ! ψ ( r ) = 0 (138)Assuming the unit radius of the billiards we get the solutions for (137) in the followingforms [27]: ˜ ψ ± ( r ) = q − e ± iλ R r q dr ′ ˜ χ ± ( r ) , r + < r ≤ q = p r = E − m λ r = Er ( r − r + )( r − r − ) , r ± = ± | m | λ √ E , m = 0 (139)which correspond to the fundamental solutions ˜ ψ + ( r ) and ˜ ψ − ( r ) defined in the respectivesectors 1 i ¯1 of the Stokes graph of Fig.13 with ˜ χ ± ( r ) normed at the corresponding sectorinfinities by ˜ χ + ( ∞ ) = ˜ χ − ( ∞ ¯1 ) = 1. The r -plane on this figure has a cut from r + to r − on which q changes its sign, while q − gets the factor ± i depending on the cut crossingdirections (+ i for crossing it upwards, − i - for downwards).41igure 14: The Stokes graph for the cylindrical billiardsThe Stokes graphs formalism corresponding to (139) easily provides us with the energyspectrum of the billiards. For this we take the wave function ˜ ψ ( r ) corresponding to this caseas a linear combination ˜ ψ ( r ) = ˜ ψ + ( r ) + c ˜ ψ − ( r ) (140)given on the segment AC where c is defined by the condition ˜ ψ (1) = 0.It is easy to see that then: c = − ˜ χ + (1)˜ χ − (1) (141)Next the combination (140) has to be continued to the segment OC vanishing on itwhen r →
0. This means however that up to a constant it should be identified with thefundamental solution ˜ ψ ( r ) = q − e − iλ R rr + q dr ′ ˜ χ ( r ), taken for example in this form abovethe cut OC attached to the sector 0 on Fig.13, i.e. for r + iǫ, < r < r + .According to the well known rules the latter can be expressed as the following linearcombination of ˜ ψ ± ( r ) [27]:˜ ψ ( r ) = − i ˜ χ → ¯1 e iλ R r + q dr ′ ˜ ψ + ( r ) + ˜ χ → e − iλ R r + q dr ′ ˜ ψ − ( r )0 < r < r + (142)where ˜ χ → j = lim r →∞ j ˜ χ ( r ) with ∞ j being the infinity point in the sector j, j = 1 , ¯1.42t then follows from (142) that:˜ ψ ( r ) = iχ → ¯1 e − iλ R r + q dr ′ ˜ ψ ( r ) (143)and c = i ˜ χ → ˜ χ → ¯1 e − iλ R r + q dr ′ (144)The equations (141) and (144) together are just the quantization conditions for the energywith the spectrum of the latter being degenerate with respect to the sign of m .To make further a correspondence with sec.5 we shall use rather the solutions ψ ± ( r ) ≡ ˜ χ ∓ (1) ˜ ψ ± ( r ), χ ± ( r ) ≡ ˜ χ ∓ (1) ˜ χ ± ( r ) and ψ ( r ) ≡ ˜ χ − (1) ˜ ψ ( r ) in the remaining considerations.Therefore instead of (140) we get simply: ψ ( r ) = ψ + ( r ) − ψ − ( r ) (145)so that and instead of (144) we have:1 = − i χ → χ ¯1 → e − iλ R r + q dr ′ (146)where χ → = ˜ χ − (1) ˜ χ → and χ ¯1 → = ˜ χ + (1) ˜ χ ¯1 → and where we have taken into accountthat χ → j = χ j → , j = 1 , ¯1.The results (141)-(146) obtained above are exact.Consider now the semiclassical limits of (141)-(146), i.e. when λ → + ∞ . It shouldbe stressed that the fundamental solution formalism allows us to do such a passage and inevery of the above formula this limit is well defined. In particular all exponentially smallcontributions have to be neglected. In this way if r is in sector 0 such an exponentially smallis the solution ˜ ψ ( r ) so that taking the semiclassical limit in (142) we get:0 = − iχ as ¯1 → e + iλ R r + q as dr ′ ψ as + ( r ) + χ as → e − iλ R r + q as dr ′ ψ as − ( r ) , < r ≤ r + ψ as ± ( r ) = q − as e ± iλ R r q as dr ′ χ as ± ( r ) (147)where ” as ” denotes the semiclassical forms of the relevant quantities and in particular χ as ± ( r ) , χ as ¯1 → (= χ as → ) and the asymptotic form of χ as → (= χ as → ¯1 ) can be found in any ofthe references [27] while q as = E − m λ r with E = E + λ E + ... .In fact the linear relation (147) is equivalent to the following identity valid in the semi-classical limit: χ as + ( r ) χ as ¯1 → = χ as → χ as − ( r ) , < r ≤ r + (148)if the point r, r ∈ OC , is approached continuing χ as + ( r ) anticlockwise and χ as − ( r ) - clockwisearound the point C of Fig.13.Taking also the semiclassical limit of the exact quantization condition (146) we get:1 = − i χ as → χ as ¯1 → e − iλ R r + q dr ′ (149)43rom both (147) and (149) it follows that: ψ as + ( r ) = ψ as − ( r ) , < r ≤ r + (150)Conversely if (150) is valid then from the identity (147) we get the semiclassical form(149) of the quantization condition.Therefore both the conditions (146) and (150) are equivalent to each other as the semi-classical quantization conditions in the case considered.We conclude therefore that the equation (150) substitutes the quantization condition (149)in the semiclassical limit, i.e. in the quantized semiclassical limit both ψ as + ( r ) and ψ as − ( r ) hasto coincide in the sector 0. But this coincidence can be analytically continued to any (nonsingular) point of the r -plane. In particular it can be continued back to the classically allowedsegment AC . These can be done in two ways, by avoiding the singular point C clockwise oranticlockwise.Before making the respective continuations let us note that the relation (142) cannot becontinued to the segment AC of the Stokes graph of Fig.13 not loosing its connection with itsasymptotic form (147) while this asymptotic form itself can loosing however its connectionto the original equation (142). This new form of an equation substituting the original one(142) can be obtained by the Borel resummation procedure.Let us start therefore with the relation (147) and (150) to continue them to AC aroundthe point C clockwise. We get:0 = − iχ as ¯1 → e + iλ R r + q as dr ′ ψ as + ( r ) + χ as → e − iλ R r + q as dr ′ ψ as,cont − ( r ) , < r ≤ r + (151)and ψ as + ( r ) = ψ as,cont − ( r ) , r + < r ≤ ψ as,cont − ( r ) denotes ψ as − ( r ) continued analytically clockwise around the point C fromthe segment AC to the same segment.If we choose the opposite path of continuation, i.e. anticlockwise one, then continuing therelation (150) we get:0 = − iχ as ¯1 → e + iλ R r + q as dr ′ ψ as,cont + ( r ) + χ as → e − iλ R r + q as dr ′ ψ as,cont − ( r ) , < r ≤ r + (153)and ψ as − ( r ) = ψ as,cont + ( r ) , r + < r ≤ ψ as,cont + ( r ) denotes ψ as + ( r ) continued analytically anticlockwise around the point C fromthe segment AC to the same segment.However, contrary to OC the segment AC lies now totally in the common part of thedomains of Borel summability of ψ ± ( r ) so that we have: (cid:0) ψ as ± ( r ) (cid:1) BS = ψ ± ( r ) , r + < r ≤ BS means the Borel resummation of the relevant quantity.But according to (152) i (154) we have instead: (cid:16) ψ as,cont ± ( r ) (cid:17) BS = ψ ∓ ( r ) r + < r ≤ (cid:16) ψ as,cont − ( r ) (cid:17) BS = i χ ¯1 → χ → e +2 iλ R r + q dr ′ ψ + ( r ) (cid:16) ψ as,cont + ( r ) (cid:17) BS = − i χ → χ ¯1 → e − iλ R r + q dr ′ ψ − ( r )0 < r ≤ r + (157)If we now apply the Borel summed quantization conditions (156) to the respective equa-tions in (157) we immediately recover the exact quantization condition (146).Let us summarize the above results. • In the semiclassical limit the quantization condition for the circular billiards is theappropriate coincidence of the asymptotic expansions of two fundamental solutions inthe classically unallowed region (the segment OC ). This coincidence is given by (150). • In the classically allowed region the semiclassical quantization condition can be obtainedby continuing appropriately (clockwise or anticlockwise) one of the semiclassical solu-tions ψ as ± ( r ) from the segment AC and back to it around the turning point C and nextby identifying the continued solution with the second one. The respective identificationsare given by (152) and (154). • The semiclassical quantization conditions formulated in the classically allowed as wellas in unallowed regions can be Borel summable leading us to the exact ones; • The fundamental semiclassical solutions continued to the classically allowed region (thesegment AC ) by the unallowed one (the segment OC ) still represent semiclassical ex-pansions of some fundamental solutions which by Borel resummation can be identifiedappropriately (see the formulae (157)). • The semiclassical quantization condition formulated by (152) and (154) in the classicallyallowed region (the segment AC ) can be Borel summable providing us with the properexact quantization conditions. B.1 The fundamental solutions semiclassical analysis rewritten on the clas-sical trajectories
Let us now rewrite the above results in the formalism of the SWF’s defined on the classicaltrajectories developed in sec.5.To this goal let us assume the semiclassical expansion (12) for the energy E so that theclassical momentum p = √ E , and associate with the solution (140) two bundle skeletons B and B A . 45oth the skeletons consist of single ray bundles only. The skeleton B consists of theray bundle B (0 , π ) the rays of which run with the definite positive angular momentum m, m = 1 , , ..., and with momenta p shown in Fig.2 while the angular momentum of theskeleton B A is equal to − m and rays of its unique bundle B A (0 , π ) have momenta p A onFig.2. For both the momenta we have | p | = | p A | = p .Any point of the ring ( r, Φ) , r + < r ≤ , ≤ Φ < π , is crossed by two rays comingfrom different bundles starting from the points s and s of the billiard boundary and havingthe respective angular momenta equal to p cos α = m/λ and p cos( π − α ) = − m/λ withcos α = r + (see Fig.2) while their corresponding radial momenta are the same and equal to p r = ˙ r = p ( t − t c ) r ( t ) = − q E − p cos αr = − q E − m λ r ≡ − q being negative ( t < t c , t c =sin α/p ).Starting at the moment t = 0 from the boundary these rays meet each other at the point( r, Φ) in the moment t .Each of the bundles defines a transformation of coordinate ( r, Φ) → ( d, s i ) , i = 1 , , by r = r ( d ) = p cos α + (sin α − d ) , d = pt and Φ = s + α − γ = s − α + γ < s + α correspondingly to the bundle, see Fig.2.The total original solution Ψ( r, Φ) = ψ ( r ) r e im Φ (158)corresponding to (145) can be now rewritten by the new coordinates noticing that R r q dr ′ = − R r p r dr ′ = − R t p r dt ′ = − R t p dt ′ + R t p r dt ′ = − p t + p cos α R t dt ′ r ( t ′ ) and that Z t dt ′ r ( t ′ ) = 1cos α arctan p ( t ′ − t c )cos α (cid:12)(cid:12)(cid:12)(cid:12) t = 1cos α ( α − γ ) (159)for t < t c so that: ψ ± ( r ( t ) , λ ) = q − e ∓ iλp t ± i | m | ( α − γ ) χ ± ( t, λ ) χ ± ( t, λ ) = (cid:18) qq (cid:19) − e ∓ iλ R t (cid:16) q − q (cid:17) q dt ′ χ ± ( r ( t ) , λ ) = (cid:18) qq (cid:19) − exp ∓ i X n ≥ E n +1 λ − n − Z d
11 + (cid:16) qq (cid:17) dt ′ χ ± ( r ( t ) , λ ) qq = 1 + P n ≥ E n λ − n − q (160)and then we get: Ψ( r, Φ) = ψ + ( r ( t )) − ψ − ( r ( t )) r ( t ) e im Φ =( r q ) − e − iλp t + i | m | ( α − γ )+ im Φ χ + ( t, λ ) − ( r q ) − e + iλp t − i | m | ( α − γ )+ im Φ χ − ( t, λ ) (161)where ( r q ) − ≡ ( r ( t ) q ( t )) − = ( E ( r ( t ) − r )) − = p − | t − t c | − .46rom (161) we get further:Ψ( r, Φ) = ( ( r q ) − e − iλp t + ims χ + ( t, λ ) − ( r q ) − e iλp t + ims χ − ( t, λ ) , m > r q ) − e − iλp t + ims χ + ( t, λ ) − ( r q ) − e iλp t + ims χ − ( t, λ ) , m < p − | t − t c | − e − iλp t − iλps cos( π − α ) χ + ( t, λ ) − p − | t − t c | − e iλp t + iλps cos α χ − ( t, λ ) , m > p − | t − t c | − e − iλp t − iλps cos α χ + ( t, λ ) − p − | t − t c | − e iλp t + iλps cos( π − α ) χ − ( t, λ ) , m < ( i Ψ − ( t, s ) − i Ψ + ( t, s ) , m > i Ψ − ( t, s ) − i Ψ + ( t, s ) , m < ± ( t, s j ) , j = 1 ,
2, have the form (26) but are exact. We have also assumed that( t − t c ) − = − i | t − t c | − for t < t c (see below).The last result shows that, depending on m , the bundles on which the fundamentalsolutions ψ ± ( r ) are defined are chosen by the latter accordingly to their signatures.Let us now consider the analytical continuation of the semiclassical limit Ψ as ( r, Φ) of thesolution Ψ( r, Φ) from the point A to its point B as it is defined by the rule (152) or back, i.e.from B to A correspondingly to the rule (154), see Fig.2.Since t = t c − p p ( r − r + )( r + r + ) then, if r rounds the turning point C by 2 π clockwisestarting from the segment AC (see Fig.13), t rounds t c on the t -plane also clockwise by π .But t rounds t c by π anticlockwise if r does the previous motion in the opposite direction.In both the cases we are found at the point B with t > t c . However in the first case( t − t c ) − = | t − t c | − while in the second ( t − t c ) − = −| t − t c | − .Ψ as ( r, Φ) continued clockwise on the r -Riemann surface is given by:Ψ asclock ( r, Φ) = ψ as + ( r ) − ψ as,cont − ( r ) r e im Φ = 1 r (cid:18) − ie − iλ R r +1 q dr ′ χ as ¯1 → χ as → (cid:19) ψ as + ( r ) e im Φ r + < r < ψ as,cont − ( r ) has been substituted by its form which follows from (153).Continued however anticlockwise it is given by:Ψ asanticlock ( r, Φ) = ψ as,cont + ( r ) − ψ as − ( r ) r e im Φ = 1 r ie +2 iλ R r +1 q dr ′ χ as → χ as ¯1 → ! ψ as − ( r ) e im Φ r + < r < ψ as,cont + ( r ) has been substituted by its form given by (151).Note that in both the above continuations we have not taken into account the quantizationconditions (152) and (154).However if the energy is to be quantized we have to have:Ψ asclock ( r, Φ) = Ψ asanticlock ( r, Φ) ≡ t, t > t c , we get from (162) and (163), see Fig.7:Ψ asclock ( r, Φ) = ( − i Ψ + ,cont ( t, s ) + i Ψ − (2 t c − t, s ) , m > − i Ψ + ,cont ( t, s ) + i Ψ − (2 t c − t, s )) , m < − ip − | t − t c | − χ as ¯1 → χ as → e iλp t + iλps cos α χ as + ( t, λ )+ p − | t − t c | − e − iλp (2 t c − t ) − iλps cos( π − α ) χ as + (2 t c − t, λ ) , m > − ip − | t − t c | − χ as ¯1 → χ as → e iλp t + iλps cos( π − α ) χ as + ( t, λ )+ p − | t − t c | − e − iλp (2 t c − t ) − iλps cos α χ as + (2 t c − t, λ ) , m < t c > t > t c , s = s ± α (166)where Ψ + ,cont ( t, s j ) , t > t c , are the results of the continuations of Ψ + ( t, s j ) , t < t c , j = 1 , asanticlock ( r, Φ) we get instead:Ψ asanticlock ( r, Φ) = ( − i Ψ + (2 t c − t, s ) + i Ψ − ,cont ( t, s ) , m > − i Ψ + (2 t c − t, s ) + i Ψ − ,cont ( t, s )) , m < − p − | t − t c | − e iλp (2 t c − t )+ iλps cos α χ as − (2 t c − t, λ )+ − ip − | t − t c | − χ as → χ as ¯1 → e − iλp t − iλps cos( π − α ) χ as − ( t, λ ) , m > − p − | t − t c | − e iλp (2 t c − t )+ iλps cos( π − α ) χ as − (2 t c − t, λ )+ − ip − | t − t c | − χ as → χ as ¯1 → e − iλp t − iλps cos α χ as − ( t, λ ) , m < t c > t > t c , s = s ± α (167)where Ψ − ,cont ( t, s j ) , t > t c , are the results of the continuations of Ψ − ( t, s j ) , t < t c , j = 1 , + ( t, s ) continued through the caus-tic has to avoid the focusing point at t = t c from above in the t -plane, moving clockwise whileΨ − ( t, s ) - from below, moving anticlockwise. By such a continuation Ψ + ( t, s ) acquires thefactor i while Ψ − ( t, s ) - the factor − i .Let us summarize the above results.1. The SWF Ψ as ( x, y ) for the circular billiards originally defined closely to the boundaryof the billiards and vanishing on the boundary can be written as the combinations (162)of the two SWF Ψ ± ( t, s ) of the opposite signatures having the forms given by (26) anddefined on the two skeletons B and B A .2. It is the SWF Ψ + ( t, s ) which is defined on the skeleton B which momenta being tan-gential to the caustic of the bundle give the anticlockwise orientation of the caustic for m > B A for m <
0. The SWF Ψ − ( t, s ) is then defined on thesecond skeleton in the respective cases.3. Ψ as ( x, y ) can be continued across the caustic avoiding the real singular point at t = t c on the t -plane from above for Ψ + ( t, s ) and from below - for Ψ − ( t, s ) and getting thenthe forms (166) or (167).4. If Ψ as ( x, y ) is to be quantized then Ψ as,cont ( x, y ) has to vanish identically independentlyof the way (clockwise or anticlokcwise) it was continued. In such a case we have to have:Ψ ± ,cont ( t, s ) = Ψ ∓ (2 t c − t, s ± α ) , t > t c (168)48eading to for t = 2 t c : χ as ¯1 → ( E, λ ) χ as → ( E, λ ) exp i X n ≥ E n +1 λ − n Z t c
11 + (cid:16) qq (cid:17) dt ′ e iλp t c − iλαp cos α + i π = 1 (169)where we have taken into account that r (2 t c − t ) = r ( t ).5. In the limit λ → ∞ the condition (169) decays into the following ones: e iλpd ( s,α ) − iλαp cos α + i π = 1 χ as ¯1 → ( E, λ ) χ as → ( E, λ ) exp i X n ≥ E n +1 λ − n Z t c
11 + (cid:16) qq (cid:17) dt ′ = 1 d ( s, α ) = 2 pt c = 2 sin α (170)Of course the condition (169) coincides with (144).6. The two degenerate SWF’s Ψ as, ± ( x, y ) in each point inside the ring cos α < r ≤ as, + ( x, y ) = − i Ψ + ( t, s ) + i Ψ + ,cont (2 t c − t, s − α ) = i Ψ − ( t, s ) − i Ψ − ,cont (2 t c − t, s + 2 α ) , m > as, − ( x, y ) = − i Ψ + ( t, s ) + i Ψ + ,cont (2 t c − t, s − α ) = − i Ψ − ,cont (2 t c − t, s + 2 α ) + i Ψ − ( t, s ) , m < t < t c (172) Appendix C
It is shown in this appendix that δ k ( u, l ) from the formula (33) and δ ( u j , l j ) from theformula (39) are s -independent. To this end consider Fig.1 on which a mapping h k ( s ; u, l ) ofthe arc A k ( u, l ) into an arc A k ′ ( u ′ , l ′ ) is defined by the the bundle B k ( u, l ). According to thismapping we have: r ( h k ( s ; u, l )) = r ( s ) + D ( s ; u, l ) r ( s ) ∈ A k ( u, l ) , r ( h k ( s ; u, l )) ∈ A k ′ ( u ′ , l ′ ) (173)where D ( s ; u, l ) is a vector linking the point r ( s ) with the point r ( h k ( s ; u, l )) of the billiardsboundary.Differentiating the equation (173) with respect to s we get: ∂h k ( s ; u, l ) ∂s cos β ( h k ( s ; u, l )) = cos β ( s ) − D ( s ; u, l ) ∂γ k ( s ; u, l ) ∂s sin γ k ( s ; u, l ) +cos γ k ( s ; u, l ) ∂D ( s ; u, l ) ∂s∂h k ( s ; u, l ) ∂s sin β ( h k ( s ; u, l )) = sin β ( s ) + D ( s ; u, l ) ∂γ k ( s ; u, l ) ∂s cos γ k ( s ; u, l ) +sin γ k ( s ; u, l ) ∂D ( s ; u, l ) ∂s (174)49here D ( s ; u, l ) is the length of D ( s ; u, l ).Making further the proper linear combinations of the last equations we have finally: ∂h k ( s ; u, l ) ∂s cos α k ′ ( h k ( s ; u, l ); u ′ , l ′ ) = cos α k ( s ; u, l ) + ∂D ( s ; u, l ) ∂s − ∂h k ( s ; u, l ) ∂s sin α k ′ ( h k ( s ; u, l ); u ′ , l ′ ) = sin α k ( s ; u, l ) − D ( s ; u, l ) ∂γ k ( s ; u, l ) ∂s (175)where we have taken into account the following relation between the angles involved: α k ′ ( h k ( s ; u, l ); u ′ , l ′ ) + α k ( s ; u, l ) = β ( h k ( s ; u, l )) − β ( s ) + 2 πγ k ( s ; u, l ) = β ( s ) + α k ( s ; u, l ) (176)which follows from Fig.1.The independence of s of δ k ( u, l ) and δ ( u j , l j ) follows now easily from the first of therelations (175). Appendix D
The billiard Laplacean expressed by the variables d, s has the following form: △ ( d, s ) = cos αJ + 1 ! ∂∂d + 1 J ∂ ∂s − αJ ∂ ∂s∂d + (cid:18) cos αJ (cid:18) ( α ′ + γ ′ ) cos α − γ ′′ γ ′ sin α (cid:19) + 1 J (cid:18) γ ′′ γ ′ cos α + α ′ sin α (cid:19) − γ ′ J (cid:19) ∂∂d − (cid:18) J (cid:18) ( α ′ + γ ′ ) cos α − γ ′′ γ ′ sin α (cid:19) + γ ′′ γ ′ J (cid:19) ∂∂sJ = sin α − γ ′ d (177)while for the corresponding operator J · △ · J − we get: J · △ · J − = 34 J − ( ∇ J ) − J − ∇ J · ∇ − J − △ J + △ ( ∇ J ) = γ ′ J (cid:16) d − d sin α + 1 (cid:17) + 2 γ ′ J ( α ′ cos α − dγ ′′ ) cos α + 1 J ( α ′ cos α − dγ ′′ ) ∇ J · ∇ = − (cid:18) γ ′ J (cid:16) d − d sin α + 1 (cid:17) + 1 J ( α ′ cos α − dγ ′′ ) cos α (cid:19) ∂∂d + (cid:18) J ( α ′ cos α − dγ ′′ ) + γ ′ cos αJ (cid:19) ∂∂s △ J = 1 J ∂ ∂s (sin α − γ ′ d ) + 2 γ ′′ cos αJ − γ ′ J (cid:16) ( α ′ cos α − dγ ′′ ) cos α − ( β ′ sin α − dγ ′ )(sin α − γ ′ d ) (cid:17) − γ ′ cos αJ ( α ′ cos α − dγ ′′ ) (178)For the circular billiards the corresponding Laplacean takes the forms: △ ( d, s ) = 1 J (cid:16) d − d sin α + 1 (cid:17) ∂ ∂d + 1 J ∂ ∂s − αJ ∂ ∂s∂d − J ∂∂d − cos αJ ∂∂sJ = sin α − d (179)50hile the corresponding operator J · △ · J − the form: J · △ · J − ( d, s ) = 34 J (cid:16) d − d sin α + 1 (cid:17) + 1 J (cid:16) d − d sin α + 1 (cid:17) ∂∂d − cos αJ ∂∂s +12 J + △ ( d, s )(180)For the rectangular billiards the corresponding Laplacean has the following two similarforms depending on the rectangular sides: △ ( d, s ) = 1sin α ∂ ∂d + ∂ ∂s ! ∓ α sin α ∂ ∂s∂d △ ( d, s ) = 1cos α ∂ ∂d + ∂ ∂s ! ∓ α cos α ∂ ∂s∂d (181)and of course J · △ · J − ≡ △ for this case of the billiards. References [1] Landau L. D., Lifshitz E. M.,
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