Formation of superscar waves in plane polygonal billiards

Polygonal billiards constitute a special class of models. Though they have zero Lyapunov exponent their classical and quantum properties are involved due to scattering on singular vertices. It is demonstrated that in the semiclassical limit multiple singular scattering on such vertices when optical boundaries of many scatters overlap leads to vanishing of quantum wave functions along straight lines built by these scatters. This phenomenon has an especially important consequence for polygonal billiards where periodic orbits (when they exist) form pencils of parallel rays restricted from the both sides by singular vertices. Due to singular scattering on boundary vertices, waves propagated inside periodic orbit pencils in the semiclassical limit tend to zero along pencil boundaries thus forming weakly interacting quasi-modes. Contrary to scars in chaotic systems the discussed quasi-modes in polygonal billiards become almost exact for high-excited states and for brevity they are designated as superscars. Many pictures of eigenfunctions for a triangular billiard and a barrier billiard which have clear superscar structures are presented in the paper. Special attention is given to the development of quantitative methods of detecting and analysing such superscars. In particular, it is demonstrated that the overlap between superscar waves associated with a fixed periodic orbit and eigenfunctions of a barrier billiard is distributed according to the Breit-Wigner distribution typical for weakly interacting quasi-modes (or doorway states). For special sub-class of rational polygonal billiards called Veech polygons where all periodic orbits can be calculated analytically it is argued and checked numerically that their eigenfunctions are fractal in the Fourier space.


Introduction
The largest part of this work has been prepared during 2003-2004 but an implacable illness of Charles Schmit had permitted to publish uniquely its short account [1]. It is only now that I collect different fragments of performed investigations and organise them in a readable form.
The paper examines the structure of eigenfunctions for a special class of quantum models, namely twodimensional polygonal billiards whose boundaries are straight lines. Classical mechanics of these problems corresponding to the ray propagation with the specular reflection from the boundaries is intricate, surprisingly rich, and notorious difficult (see, e.g., [2] and references therein).
The most investigated is the case of rational or pseudo-integrable billiards where all (internal) billiard angles q j are rational fractions of π ( ) q p = m n 1 j j j with co-prime integers m j and n j . A characteristic property of such billiards is that their classical trajectories belong to 2-dimensional surfaces of finite genus g related with angles (1) as follows (see, e.g., [3]) where N is the least common multiple of all n j .
Two particular examples of such models discussed in the paper are depicted in figure 1. The first model is a right triangular billiard with angles [ ] p p p 8, 3 8, 2 . The second one is a rectangular billiard of sides a 2 and b with a barrier of height b 2 in the middle. This model has 6 corners with angles p 2 and one conner with angle 2π.
In a billiard where all angle numerators m j =1 trajectories belong to tori with g=1 and the model is classically integrable. The list of 2-dimensional integrable polygonal billiards is limited. It  If at least one numerator is bigger than 1, classical trajectories lie on a surface of genus  g 2 and such models are genuine pseudo-integrable models. The billiards at figure 1 have genus g=2. The values of genus can be obtained by the explicit unfolding of the initial billiard table. For example, at figure 2 the unfolding of the right triangular billiard with angle p 8 is performed. By reflections one gets a surface with the shape of the regular octagon whose opposite parallel sides are identified. Topologically the resulting surface is a sphere with 2 handles which is the canonical image of genus 2 surfaces.
The quantisation of billiards consists in finding eigenvalues and eigenfunctions of the wave equation For all integrable polygonal billiards cited above the solution of the quantum problem is well known (see e.g. [3]). Their eigenvalues and eigenfunctions depend on two integers. Eigenvalues are quadratic functions of these integers and eigenfunctions are finite combination of trigonometric functions. Even the inverse theorem is valid:  the list of billiards whose all eigenfunctions are finite combinations of trigonometric functions is exhausted by the above integrable billiards [4].
The structure of eigenvalues and eigenfunctions of polygonal billiards are much more complicated and only partial results are available.
In quantum chaos there are two big conjectures concerning the spectral statistics of generic integrable and fully chaotic systems. For integrable models the spectral statistics (after unfolding) coincides with the Poisson statistics of independent random variables [5] and for chaotic systems it corresponds to the eigenvalue statistics of the standard random matrix ensembles depended only on symmetries [6]. The difference between these two types of universal statistics is clearly seen in the behaviour of the nearest-neighbour distribution, p(s), which gives the probability that two nearest levels are separated by a distance s (see e.g. [7,8]  The numerical calculations of pseudo-integrable billiards [3], [9][10][11][12][13][14][15][16][17][18] demonstrate that their spectral properties are in-between these two universal distributions. Namely, their nearest-neighbour distribution reveals a linear level repulsion ( ) ⟶  p s s s 0 as for the random matrix ensemble with b = 1 but at large distances p(s) decreases exponentially as for the Poisson statistics. The purpose of this paper is to investigate certain properties of eigenfunctions for plane polygonal billiards. The principal difficulty in treating such problems is the fact that in polygonal billiards the vertices with angles p ¹ n with integer n are singular points for the classical motion. If a parallel pencil of rays hits such vertex it splits discontinuously into two different pencils (cf figure 3). Quantum mechanics has to smooth these singularities and leads to the notion of singular diffraction. The exact solution for the scattering on wedge has been obtained long time ago by Sommerfeld [19] (cf also [20]). The simplest case of such diffraction corresponds to the scattering on a half-plane with, e.g., the Dirichlet boundary conditions, see figure 4(a).
The exact solution for this problem has been found by Sommerfeld in 1896 [19] and in polar coordinates it reads From the expansion of ( )  Y r at large distances one finds that the total wave splits into two contributions, the incident plane wave and the out-going cylindrical wave Sommerfeld [19] also found the exact solution for the scattering on an arbitrary wedge with the Dirichlet boundary conditions as at figure 4(c). In this case the diffraction coefficient has the following form and α is the wedge angle. The main feature of such diffraction coefficients is the existence of certain lines where diffraction coefficients formally blow-up. These lines are called optical boundaries and they correspond to zeros of the denominators in the above formulas. For the scattering on a half-plan they appear when Physically these lines separate regions with different numbers of geometrical rays and are a manifestation of the discontinuous character of classical ray motion, cf figure 4(b). As in quantum mechanics wave fields are continuous, the separation of the exact field into a sum of the free motion (plane wave) plus a small reflected field is not possible in a vicinity of such optical boundaries which forces the diffraction coefficient to diverge. Consequently, the diffractive coefficient description cannot be applied in parabolic regions near optical boundaries where the dimensionless arguments of the F-functions are of the order of 1, =d j u krsin 1 2 , and dj is the angle of deviation from optical boundaries, cf figure 4(b).
Difficult problems appear when inside these intermediate regions there are other points of singular diffractions which is inevitable for plane polygonal billiards. For a finite number of singular diffraction vertices it is possible to develop the uniform approximations which give a good description of the multiple singular diffraction in the semiclassical limit  ¥ k (see, e.g., [21] and references therein). For an infinite number of singular diffractions the situation is less clear. To understand the behaviour of waves scattered on many singular scatters where optical boundaries strongly overlap three interrelated approaches are discussed in section 2. All these methods demonstrate that the multiple singular diffraction in the semiclassical limit of high energy scattering leads to a non-perturbative effect of (almost) vanishing of eigenfunctions along straight lines passed through singular scatters (vertices with angles p ¹ n). Consequently, a wave scattered with a small incident angle from many singular scatters arranged along a line will be reflected from them as from a mirror with the Dirichlet boundary condition although the mirror itself does not exist. The importance of this phenomenon for polygonal billiards is related with the fact that periodic orbits in such billiards (when they exist) form families of parallel trajectories, cf figure 6(c). When unfolded each family constitutes an infinite pencil (or a channel) restricted from the both sides by singular scatters. Such configuration is exactly the one which permits the propagation of a plane wave with (approximately) the Dirichlet boundary conditions along two fictitious mirrors built by singular scatters. The validity of such approximation becomes better in the semiclassical limit of high energy. Therefore we propose to call these waves ʼsuperscars' to distinguish them from the scar phenomena in chaotic systems [22][23][24] where individual scar amplitudes decrease in the semiclassical limit. Numerous examples of numerically computed high-excited eigenfunctions with clear superscar structures for the triangular and the barrier billiards depicted at figure 1 are presented in section 3. An additional confirmation of the applicability of the superscar picture is the very good agreement of the true eigen-energies of such states with the superscar energies computed analytically from the knowledge of periodic orbit parameters.
To get a more quantitative information about the formation of superscar waves the overlaps between consecutive barrier billiard eigenfunctions and specific folded superscar waves are investigated in section 4. It is observed that in a small vicinity of all superscar energies there exist true eigenstates having large overlaps with the corresponding superscar wave. In a finite energy window the values of the overlap fluctuate according to the Breit-Wigner distribution whose parameters agree with the ones calculated analytically in section 2. Another useful approach discussed in the same section is the Fourier-type expansion method. It consists in the expansion of true eigenfunctions in a series of convenient basis functions. The existence of superscars manifests as anomalously large values of certain expansion coefficients.
If a periodic orbit family exits in a given polygonal billiard it may and will support superscar waves. But for generic polygonal billiards little is known about periodic orbits. Only for a special sub-class of pseudo-integrable billiards called Veech polygons [25] one can find all periodic orbits analytically. Billiards considered in the paper belong to this class. For Veech polygons it is possible to calculate analytically the level compressibility [15,18] which is practically the only one spectral characteristic accessible to analytical calculations. It is believed (and confirmed numerically for many different models, see e.g. [26]) that systems with non-trivial compressibility should have eigenfunctions with non-trivial fractal dimensions. For pseudo-integrable billiards the above mentioned strong fluctuations of Fourier coefficients mean that eigenfunctions in the momentum space may have fractal dimensions. In section 5 it is numerically demonstrated that indeed eigenfunctions of the barrier billiard do have non-trivial fractal dimensions. Section 6 contains a brief summary of obtained results. Appendix is devoted to the investigation of periodic orbit pencils in the barrier billiards and the folding of corresponding superscar waves.

Singular multiple diffraction
The purpose of this section is to present different approaches to the multiple singular scattering on a periodic array of singular vertices (wedges with angles p ¹ n with integer n) arranged along a straight line as indicated at figure 5. The simplest method consists in the construction of the Kirchhoff-type approximation to this problem. It has been done in [21] and briefly reviewed in section 2.1. It is known that the condition of applicability of the Kirchhoff approximation is not easy to be rigorously established. To get more precise information of this process, the exact solution for the scattering on staggered periodic set of half-planes as indicated at figure 6(a) derived by Carlson and Heins in 1947 [27] and analysed in the semiclassical limit in [28] is discussed in section 2.2. Section 2.3 is devoted to numerical investigation of wave propagation inside periodic array of slits depicted at figure 6 The main result established in that sections is the fact that the small-angle high-energy multiple scattering on singular wedges is equivalent to the much simpler specular (i.e. mirror) reflection from a straight line passing through the wedge apexes though the line itself does not constitute a physical boundary. In section 2.4 it is demonstrated that this result applied to polygonal billiards proofs the existence of special weekly interacting quasi-modes corresponding to plane waves propagating inside periodic orbit channels (when they exist). These quasi-modes called in the paper superscars are a specific feature of polygonal billiards. They do not exist neither in integrable nor in chaotic systems and are a non-perturbative consequence of the multiple singular diffraction inherent for polygonal billiards.

The Kirchhoff approximation
A direct approach to multiple singular scattering consists in the construction of the uniform approximation based on the Kirchhoff approach (see e.g. [19]) which corresponds to the calculation of semiclassical contributions of piecewise linear trajectories indicated at figure 5. In this approximation the role of wedges is reduced to the restriction of integration domains to half-lines ( ) ¥ 0, (cf figure 5). This problem has been investigated in [21] where it has been proved that the contribution to the trace formula from such trajectories for a finite number (n) of wedges (i.e. ( ) + n 1 -fold integral over all x j at figure 5) can be calculated analytically and the result is For large numbers of scatters the sum over q can be substituted by the integral and [28] It has been demonstrated in [28] that this result for the multiple scattering on the periodic set of wedges is equivalent to the specular reflection (with the Dirichlet boundary condition) of the incident wave from a straight (fictitious) mirror which passes through the apexes of all wedges. For small incident angle j with respect to that (fictitious) mirror the effective reflection coefficient for high-energy scattering determined from equation (12) is the following Heuristically this result can be understood as follows. By construction, the exact wave function for the scattering on a system of wedges (with the Dirichlet boundary conditions) equals zero at wedge apexes. When many wedges are aligned and the incident angle is small the visible distances between apexes are also small and in the semiclassical limit this discrete set of points can be approximated by a straight line which explains the dominance of specular scattering with the Dirichlet condition.

Scattering on staggered periodic set of half-planes
Though the Kirchhoff approximation discusses in the preceding section does indicate that the multiple singular diffraction leads to an effective scattering from a (fictitious) mirror formed by singular scatters it is difficult, in general, to prove rigorously the applicability of this approximation. In this section an exact solution of a similar problem of scattering of a plane wave with incident angle j k z x inc i cos sin on a periodic set of half-planes separated by perpendicular distance a is discussed. The apexes of all half-planes belong to a straight line and planes are inclined with respect to this line by angle α (cf figure 6(a)). The field at large distances is the sum of the reflected (into the upper half-plane) and transmitted (into the lower half-plane) fields. The total reflected field is the sum of finite number of reflected plane waves where R n are the reflection coefficients and j n are the reflected angles determined due to the periodicity from the grating equation  Here d is the distance between the apexes of half-planes, a = d a cos , and p = Q kd is the dimensionless momentum.
The total transmitted field is the same as inside straight tubes built by half-planes with the Dirichlet boundary conditions and T m are transmission coefficients. It has been shown in [27] that the above problem is soluble by the Wiener-Hopf method but the calculations in that article were performed only for low values of momenta. In [28] this problem has been reconsidered in the semiclassical limit  ¥ Q and it was demonstrated that in that limit infinite products inherent in the Wiener-Hopf method and, consequently, reflection and transmission coefficients can be obtained analytically. The most difficult (and the most interesting for us) is the case of the small-angle scattering when the incident angle j  0 as within the optical boundary of one scatter there exist many other scatters.
The main conclusions of [28] for this problem in the limit  j Q 1 and  ¥ Q are as follows: • The reflection coefficient with n=0 in equation (16) corresponding to the specular (mirror-like) reflection, Notice that this expression coincides with equation (13) obtained in the Kirchhoff approximation.
• Reflection coefficients when > n 0 in equation (16) is kept fixed and  ¥ Q corresponding to small reflection angle, j » n Q 2 • When p a p < < 2 the transmission is negligible and the large-angle reflection coefficients dominate when n is close to a = n Q sin 2 * and j n is near to p a -2 2 (as for the specular reflection from the full inclined plane). For small j these coefficients are proportional to j 2 the large-angle reflection coefficients are negligible and the transmission coefficients are with the same functions r(u) and ( ) a u n as in (20) and (21).
The main conclusion from the above expressions is that for the sliding-type multiple scattering when the incident angle is small,  j Q 1, and  ¥ Q the dominant contribution to the reflected field comes only from one term with n=0 ( » -R 1 0 and j j = n ). This term corresponds to the specular reflection from a fictitious mirror built from a straight line passing through singular scatters (indicated by dashed lines at figure 6) The formation of a quasi-mirror boundary where in the semiclassical limit the total field tends to zero is a nonperturbative effect of the small-angle multiple scattering on singular scatters (i.e. vertices with angle p ¹ n).
The existence of the exact solution permits also to find a small leakage of the specular reflected wave after one scattering into other channels. The modulus of the amplitude of that wave deviates from unity by a small amount (when  j Q 1) as follows Scattering on a periodic array of slits To investigate this phenomenon further it is instructive to investigate the propagation of waves inside periodic array of slits with Dirichlet boundary conditions as indicated at figure 6(b). The problem corresponds to finding the solution of the Helmholtz equation which vanishes at indicated slits and is generated by the plane wave along z-axis.
In the Kirchhoff approximation (see e.g. [19]) the wave ( ) by the following relation valid provided the width w is much smaller than the distance between slits d Periodicity of the slits requires that ( ) 0 , where λ determines the propagation and attenuation due to scattering on slits. Therefore the considered problem is reduced to the following equation No analytical solution of this (simplified) equation is known. Nevertheless, as it has been discussed above, in the semiclassical limit  ¥ k its solution should be close to waves propagating inside a rectangular slab restricted by straight lines passing through corners of the slits (denoted by dashed lines at figure 6 To check this statement the numerical calculation of equation (27) was performed. To simplify numerics all space variables were rescaled in units of w 2 and the Wick rotation has been performed. It leads to a simpler equation x y n nn 1 Eq. (29) is the Fredholm integral equation of the first kind with symmetric kernel and it has a discrete set of eigenvalues L n (L L ¼   , 1 2 ) and eigenfunctions ( ) Y x n which were determined numerically. Due to the symmetry solutions are either even or odd with respect to coordinate inversion: The main conclusion of these (and others) calculations is that in the semiclassical limit k  ¥ eigenfunctions of (29) are indeed well described by the simple waves as in equation (28). An important characteristics of such waves is the requirement that they vanish as k  ¥ at boundaries of effective slab which implies the quantisation of transverse momentum. At figure 8(b) the values of true eigenfunctions at the boundary, ( ) Y 1 n , are plotted. It is clearly seen that with fixed p n and with increasing of κ this value indeed tends to zero. It was observed that these values are well described by the following asymptotic formula where the constant » C 1.65 is numerically the same as in equation (25), The equality of these two constants can be explained as follows. The appearance of the imaginary part of the wave energy physically means that propagating wave escapes into other channels. The modulus squared of this wave after passing the distance L decreases by dE L k Im . According to equation (25) after each collision with slits this quantity decreases by j C kd . When a wave propagates with the incident angle j along the distance L it has approximately ( ) j L w collisions. Therefore the total leakage is As j » p k n one reproduces the imaginary part of equation (33).

Application to polygonal billiards
The multiple scattering on singular wedges with a p ¹ n is, in general, a complicated problem, especially when optical boundaries of different scatters overlap. The above discussion proofs that in the semiclassical limit when singular scatters are arranged along a straight line and the incident wave is inclined with a small angle with respect to this line the reflected wave dominates by a specular reflection from that line though the line itself does not constitute a physical boundary. The Kirchhoff approximation discussed in section 2.1 clearly demonstrates that this result is independent of wedge shapes. Such non-perturbative effect is especially important for polygonal billiards where classical periodic orbits appear in families which after unfolding form infinite periodic pencils (or channels) limited from the both sides by singular vertices, cf figure 2. Consider one pencil corresponding to a primitive periodic orbit with period L p and let w be its width, see figure 6(c). Of course, the horizontal pencil boundaries do not exist but they are constituted by singular scatters. Due to multiple singular diffraction a wave propagating inside such pencil approximately vanishes at effective horizontal boundaries and therefore will take the form of a plane wave as in equation (28) ( The energy of such wave is It is plain that such wave is only an approximation to (a much more complicated) exact solution. The validity of this approximation is governed by the dimensionless perturbation parameter l j p = kL p where j is the angle between the wave direction and the horizontal boundaries. For the plane wave (35) with a small transverse momentum j » p k, p = p n w, and p » = k q m L p . Therefore the wave (35) will be a good approximation provided the following dimensionless parameter is small (the smaller the better) The values of integer n are also restricted The requirement that  n 1 leads to the conclusion that at fixed energy not all periodic orbit pencils can support propagating waves. As where A is the billiard area and ( ) g =  1 is the fraction of the billiard area covered by a given family of periodic trajectories, the length of a propagating channel is restricted as follows Long-period channels with > L L p max are closed and cannot support propagating waves. An important property of discussed propagating waves is that they become more visible (i.e., more isolated from other states) when the parameter (38) is decreasing. But for a given periodic orbit (i.e., fixed L p and w) when transverse momentum p (i.e., n) is kept fixed but energy (i.e., m) increases this parameter goes to zero. Consequently in the semiclassical limit any periodic orbit pencil may and will support such propagating quasimodes. That phenomenon resembles the formation of scars around of unstable periodic orbits in chaotic systems [22,23,29] but contrary to the usual scars the discussed quasi-modes become practically exact in the semiclassical limit. It explains the name superscars proposed for these quasi-modes. In the next section many examples of such superscars are presented for the billiards depicted at figure 1.
The problem considered in this section looks similar to the bouncing ball scarring in the stadium billiard (see, e.g., [30]) but the above arguments cannot be applied to the bouncing ball case. The discussed nonperturbative superscar formation is based on the existence of strong diffraction on singular vertices, i.e., points where classical ray dynamics is discontinuous. But the tangent to the stadium boundary is continuous along the boundary. Therefore classical ray dynamics in the stadium is also continuous and singular vertices are absent. The bouncing ball pencil in the stadium billiard is restricted by vertices with the boundary discontinuity only in the second derivative which generate merely a conventional diffraction whose analysis is beyond the scope of the paper.

Examples of superscars in triangular and barrier billiards
Consider the billiard in the shape of the right triangle with one angle equals p 8 as at figure 1(a). One of the simplest periodic orbit family of this billiard corresponds to trajectories perpendicular to the both catheti as indicated at figure 2(b). When unfolded this family fills the rectangular pencil shown at figure 2(c). The length of this rectangle (i.e., the periodic orbit length) equals twice the length of the largest cathetus and its width is the length of the smallest cathetus. According to the above discussed multiple scattering on singular points the superscar wave should propagate inside this rectangle with the Dirichlet boundary conditions on horizontal boundaries. As vertical boundaries are a part of the triangle boundaries the wave have to vanish on these boundaries as well. Taking into account symmetry of the problem one concludes that the unfolded superscar wave in the semiclassical limit obeys the Dirichlet boundary conditions on all sides of the rectangle indicated at figure 9 and has the form where a, b are lengths of respectively the largest and the smallest cathetus ( are introduced to stress that this expression exists only inside the rectangle. The energy of such state is According to the previous section such superscar approximation is valid provided that the perturbation parameter (38) (with = L a p and w=b) is restricted. To emphasise the importance of this parameter, its value for the corresponding superscar wave is indicated in the captions of eigenfunction figures below.
The superscar wave is simple (cf equation (41)) only after unfolding. When folding inside the original triangle it takes the following form is given by equation (41) (with Θ-functions included). To examine the correspondence between (approximate) superscar waves and the true quantum eigenfunctions numerical calculations of high-excited states were performed. The area of the billiard is normalised to 4π in order that the mean distance between consecutive high-energy levels equals 1. To find what numerically calculated (true) eigenfunctions resemble to superscar waves the following procedure has been used. First, values of integers m and n with  n m were chosen and the superscar energy was calculated from equation (42). Then from numerically calculated eigenvalues the one closest to the superscar energy has been selected. In all investigated cases the corresponding eigenfunction reveals clear picture very similar to the folded superscar wave (43). ) 201, 1 are presented. These eigenfunctions clearly have the same structure that the superscar waves and the indicated exact energies agree well with superstar energies calculated from equation (42). Figure 10-12 clearly validate the formation of superscar states around the simplest periodic orbit pencil in the triangular billiard. But even that orbit requires 5 scatterings from the boundary (cf figure 2(b)) and the folded superscar function is complicated (cf figure 10(a)). Longer periodic orbit pencils will necessarily be more elaborate and, consequently, the structure of corresponding superscar functions would be less clear.
To visualise better superscar structures, it is convenient to investigate the barrier billiard as at figure 1(b) where short-period orbits are simpler (see below). In numerical calculations only symmetric modes of this billiard were considered. Now the problem is reduced to the solving the Helmholtz equation , 0at all boundaries of the desymmetrised rectangle indicated at figure 1(c) except the segment = < < 0 . In calculations the aspect ratio of the barrier billiard, b/a, is chosen equal to + » 5 1 1.8and the area of the billiard is normalised to 4π. A bunch of high excited eigenfunctions around the 10000 th level for this billiard was obtained numerically and eigenfunctions corresponding to a few superscar waves were selected as it has been discussed above. For clarity at certain figures below nodal domains of these eigenfunctions were plotted. Black (white) regions correspond to points where ( ) 0respectively. At other figures it was more convenient to show grey images of the eigenfunction modulus.   2) and the second is associated with the motion between the Dirichlet and Neumann boundaries ( ), cf figure 15. The superscar waves propagating inside these two pencils are  propagating between the left part of the billiard as it should be for the Dirichlet-Dirichlet vertical bouncing ball (cf figure 15(a) and (46)). The second part is built from irregular waves with much smaller amplitudes. If superscar picture would be exact, this part should be exactly zero but as the superscar wave is only an approximation such regions have to be constituted of small-amplitude waves with irregular nodal domains. Such co-existence of two different parts of eigenfunctions is typical for superscar waves propagating in pencils with odd n a (see bellow). The calculated energy of the corresponding superscar with m=85 and n=1 =  10209.65 85,1 DD is very close to the exact energy. With the increasing of the perpendicular momentum n superscar waves become less pronounced as the parameter (38) which controls the validity of superscar approximation grows. Nevertheless the vertical bouncing ball structure remains visible even for n=6 and m=84 as shown at figure 16(b). Notice that the   2 . When folding back to the original barrier billiard this orbit gives rise to two periodic orbit pencils as shown at figure 19(b). In the usual rectangular billiard these two pencils may be continuously transformed one into another but in the barrier billiard they are restricted by singular vertices and constitute two different pencils which should be treated separately. The superscar waves propagating in the pencils have energies given by the expression The existence of different pencils manifests in different phases accumulated by a wave when propagating inside the pencils. It is plain that even and odd m correspond to the pencils indicated at figure 19(b). At figure 20 and 21 certain examples of superscar waves associated with the ( ) -1 1 orbit are presented. The first figure corresponds to m=348 and m=347 with n=1 and the second one shows eigenfunctions with the same values of m but n=2. The effect of switching from one pencil to another for even and odd m is clear visible.

Quantitative characteristics of superscars
Numerous pictures of the superscar waves formation were presented in the previous section. But such pictures are useful only to illustrate a few superscar waves associated with short-period orbit pencils. To get a quantitative information about the whole structure of eigenfunctions in plane polygonal billiards it is convenient to calculate numerically the overlap between an exact eigenfunction with energy l E and a superscar wave propagated in a fixed periodic orbit pencil x y x y x y , , d d .   In calculations the transverse quantum number n (which exists only due to the singular diffraction) is kept fixed but the longitudinal quantum number m (denoted below by m(E)) has been adjusted for different energies E in such a way that the energy difference where [ ] x denotes the integer closest to x. A technical difficulty in this approach is the calculation of the folded superscar wave, . The superscar wave is simple when it is unfolded. Due to the folding back of periodic orbit pencils, superscar waves inside the original billiard become complicated. For simple orbits the folded wave can be directly calculated as it has been done in equation (43). In Appendix it is shown how to calculate folded superscar function associated with an arbitrary periodic orbit pencil.
The overlaps between all eigenstates in the interval < < l E 2000 4000 and all superscar waves propagating inside the ( ) -0 1 pencil (horizontal bouncing ball), the ( ) -1 0 pencil (left vertical bouncing ball), and the ( ) -1 1 pencil are presented at figure 28(a)-30(a). Each of these figures shows the overlap for the four lowest transverse quantum numbers, = ¼ n 1, ,4. Every time when the energy of a true eigenstate is close to the superscar energy the corresponding eigenfunction has a considerable overlap with the superscar wave. As expected, small n leads to higher values of the overlap. To analyse quantitatively the structure of overlap peaks it is instructive to calculate their local density for each fixed n defined as follows (dE is the difference between the true energy l E and the best superscar energy where the averaging is taken over all peaks in a given energy interval where  e E. For = n 1, 2, 3, 4 this local density is plotted at figure 28(b)-30(b).
As has been discussed above superscar waves can be considered as long-lived states which interact weekly due to residual interactions governed by parameter (38). From general considerations [31][32][33][34][35], it is expected that in such situation the local density should be well approximated by the Breit-Wigner distribution Here w is the width of a periodic orbit pencil and d is the distance between singular vertices along the pencil boundary. where q and k are integers. As the basis trigonometric functions in the right-hand part of this expansion are orthogonal inside the rectangle ( ) a b 2 , the expansion coefficients F q k , can be calculated by the inverse Fourier transform  For the desymmetrised barrier billiard as at figure 1(c) no preferential system of expansion exists. Expansion (58) inside the desymmetrised barrier billiard gives rise to two different series depended on the parity of k. For odd  It means that the energy conservation forces coefficients in these figures to be close to a quarter to the ellipse. Noticeable exception is seen at figure 31(b) where certain coefficients deviate considerably from the constant energy curve. It is plain that it corresponds to the above mentioned Gibbs phenomenon. If this eigenfunction is expanded into odd series (61) such large deviations would disappear. But for orbits with even M like the ( ) -2 1 orbit indicated at figure 22(a) the expansion coefficients have Gibbs tails which could not be removed by a simple change of the basis (cf figure (32)(a)).
It is clear that well isolated superscar states associated with short-period orbits are rare. Typical eigenfunctions may contain a certain number of large coefficients corresponded to a kind of superposition of many different superscar waves (see figure 32(b)).

Fractal dimensions
Everything discussed in the previous sections about a superscar wave propagating inside a periodic orbit pencil could be applied to an arbitrary polygonal billiards (even with irrational with π angles) where there exists at least one periodic orbit family. Unfortunately periodic orbits in generic polygonal billiards is an elusive subject and even the existence of one periodic orbit, in general, is not guaranteed. Only for a special sub-class of rational polygonal billiards called Veech polygons [25] where there exits a hidden group structure one can control all classical periodic orbits.
The knowledge of periodic orbits in such models permits to calculate analytically an important characteristic of their spectral statistics, namely the spectral compressibility, χ, [15,17] which determines the linear growth of the number variance with the length of the interval Here n(L) is the number of energy levels in an interval L,¯( ) n L is the mean number of levels in this interval normalised to unit density,¯( ) = n L L, and the average is taken over different intervals of length L in a small energy window. For the Poisson distribution typical for spectral statistics of integrable systems c = 1 and for the standard random matrix ensembles which describe spectral statistics of chaotic systems c = 0. The right triangular billiard with p 8 angle has c = 5 9 [15] and the barrier billiard considered in the paper has c = 1 2 [17].
Spectral statistics of models with non-trivial compressibility, c < < 0 1, are called intermediate statistics. Many pseudo-integrable billiards belongs to this class [3], [9][10][11][12][13][14][15][16][17]. The characteristic features of the intermediate statistics are (i) a level repulsion on small distances as for the usual random matrix ensembles, (ii) an exponential decrease of the nearest-neighbour distribution on large distances similar to the Poisson distribution. These properties have been observed in numerical calculations but have not been fully proved mathematically. Numerics (and certain analytical arguments [26]) also suggest that for models with intermediate spectral statistics eigenfunctions are fractal (in general, even multifractal). The notion of multifractality (see, e.g., [36,37] and references therein) is related with a natural question about the number of important components in eigenfunctions. Let an eigenfunction with eigenvalue E be written as an expansion in a certain basis Here  is the total number of components. The central question in the multifractal formalism is the scaling of the moments of expansion coefficients with  . Define the moments with arbitrary q as follows The inverse of these moments, , is called the participation ratio. The multifractality means that moments of eigenfunctions (or their inverse) scale as a certain power of total number of wave function components where D q are called generalised fractal dimensions.
If only a finite number of components gives contributions to an eigenfunction (66) (which means that the state is localised) then D q =0. In the opposite case of completely extended states when all components are of the same order then from the normalisation it follows thatñ - A 1 2 and consequently D q =1. In [38] the multifractality was observed in the 3-dimensional Anderson model at the metal-insulator transition and later it has been investigated in different matrix models [37]. For systems with non-trivial spectral compressibility, c < < 0 1, numerical and partly analytical calculations [26] suggest that fractal dimensions in the Fourier space should be also non-trivial, < < D 0 1 q . To check these predictions numerical calculations of fractal dimensions were performed for high-excited states in the barrier billiard. Each eigenfunction has been expanded into the Fourier series (60) were computed. At figure 33(left) the participation ratio R 2 for 3 energy intervals close to the 1000 th , the 4000 th , and the 10000 th levels for the barrier billiard are presented. For comparison at this figure the same quantity but for the quarter of the (chaotic) stadium billiard with the same aspect ratio are shown for comparison. The area of the both billiards is 4πand the energies approximately equal the level numbers.
At figure 34(a) these data were used to calculate average values of the participation ratios for the barrier billiard and the stadium billiard. As expected, for the chaotic stadium billiard participation ratio scales linear with the momentum. The best fit gives ( ) = R E k 0. . Therefore, these results suggest that » » D D 0.5 2 3 . Of course, much more calculations should be done to establish correct values of fractal dimensions for pseudo-integrable billiards.

Summary
Wave functions are on the very basis of quantum mechanical calculations and the investigation of their properties are important for many applications. For generic classically chaotic systems Berry's conjecture    [39,40] stipules that in the semiclassical limit typical quantum eigenfunctions are random superpositions of elementary functions with fixed momentum whose coefficients are independent Gaussians random variables with zero mean and variance determined from the normalisation. Nevertheless, it does not signify that all chaotic eigenfunctions are completely structureless (cf [41]). It is well known that eigenfunctions in certain models may have structures (called scars) in a vicinity of unstable periodic orbits [22,23,29]. Contributions of individual unstable periodic orbits decrease with increasing of energy and a general belief is that the scar phenomenon in chaotic systems will be suppressed (or even disappear) in the semiclassical limit though a certain increase of amplitudes could still be detected [24].
The main message of this paper is that eigenfunctions of plane polygonal billiards have clear structures associated with periodic orbit families. The mechanism of formation of such structures is not periodic orbits themselves but the singular scattering on billiards corners whose angles p ¹ n with integer n. Classical ray scattering on such scatters are discontinuous but quantum mechanics substitutes singularities by formally smooth regions (called optical boundaries) where wave function changes so quickly that the separation of the wave into the incident and scattering fields is not possible. When optical boundaries of many scatters overlap the result in the semiclassical limit corresponds to vanishing of eigenfunctions along straight lines formed by singular scatters. As periodic orbits in polygonal billiards form parallel families (pencils) restricted by singular scatters such pencils may and will support propagating waves reflected from pencil boundaries passing through singular vertices as from mirrors. The validity of such specular reflection from fictitious mirrors becomes exact in strict semiclassical limit  ¥ k which explains the existence of such structures at very high energies contrary to scars in chaotic systems. To stress this fact we propose to call them ʼsuperscars'. Many pictures of superscars in simple billiards are presented in the main part of the paper. For the barrier billiard certain superscar waves were observed in microwave experiments [42,43].
In principle, superscar waves may exist in any polygonal billiards. But there is no theorem that guarantees the existence of even one classical periodic orbit for generic polygonal billiards. Consequently the superscar construction could be applied for polygonal billiards with at least one periodic orbit family. To construct a superscar wave associated with a periodic orbit pencil it is necessary to know the periodic orbit length, L p and the width of the pencil, w, restricted by singular billiard corners. The unfolded superscar wave has the form of the plane wave propagating inside the pencil as in equation (A3) and its energy is the same as for a wave propagating inside rectangular slab with the Dirichlet boundary conditions: 2 . The parity of longitudinal quantum number m is determined by the total phase accumulated by the periodic orbit. The transverse quantum number n can be arbitrary but  n m. The simplest verification of this construction consists in computing a few states with energies in a vicinity of the superscar energy with different m and a fixed n. Numerical calculations performed in the paper show that in a small vicinity of superscar energies there always exist true eigenstates which have the clear structure related with the folded superscar wave. To get quantitative view of such phenomenon it is useful to calculate numerically the overlap between true eigenfunctions and the folded superscar wave. At least for the barrier billiard such overlap has the expected Breit-Wigner form (56) whose parameters agree with analytical estimates.
Further progress depends on the possibility to control periodic orbits in polygonal billiards which is a complicated problem. Only for a special sub-class of polygonal billiards called Veech polygons [15,25] one can find analytically all periodic orbits and their parameters. The right triangular billiard with one angle p n and the barrier billiard considered in the paper belong to this class. For Veech polygons one can argue that eigenfunctions in the momentum representation are fractal with non-trivial fractal dimensions which is confirmed by numerical calculations for the barrier billiard.
The investigations presented in the paper clearly demonstrate that eigenfunctions of polygonal billiards and especially of pseudo-integrable ones have interesting and unusual properties different from the both integrable and chaotic systems. In the absence of mathematical theorems more detailed examinations of such phenomena are highly desirable.

Appendix A. Properties of periodic orbit pencils and superscar waves in rectangular billiards
The purpose of this appendix is to present (without proofs) main properties of periodic orbit pencils (POP) and the corresponding superscar waves (SW) for the rectangular billiard and to propose a method of the visualisation of folded supescar waves.
• A rectangular billiard with sides a and b cover the whole plane under the group of reflections at its boundary.
This group consists of 4 inversions:     x x y y , and integer translations:  +  + Î  x x ma y y nb m n 2 , 2 , , .
• A non-oriented primitive periodic orbit in the folded rectangular lattice can be represented by a line which connects the origin with a point with coordinates Ma 2 and Nb 2 (here and below the coordinate system is chosen as at figure 1(c)) where M and N are positive co-prime integers (peculiarities for zero M or N are easy to take into account). Here [ ] z is the largest integer lower than z,   Here ( ) Q x is the Heaviside function ( ( ) Q = x 1 for > x 0 and ( ) Q = x 0 for < x 0) introduced to stress that superscar waves are zero outside the indicated intervals. In the above formulas it is implicitly assume that x and y in the definitions (A1) have to be calculated from the central vertices of the initial rectangle. For the coordinate system as at figure 1(c) they correspond to the substitution x x x c , y y y c .
• If the symmetric expressions for the superscar waves (A10)-(A16) are used only 2 images should be taken into account and x y x w e x y x y , , ,