Spectral Form Factors of Rectangle Billiards

Transcription

1 Commun. Math. Phys. 99, 69 (998) Communications in Mathematical Physics Springer-Verlag 998 Spectral Form Factors of Rectangle Billiards Jens Marklof, Division de Physique Théorique, Institut de Physique Nucléaire, F-946 Orsay Cedex, France, and Institut des Hautes Études Scientifique, 35, Route de Chartres, F-944 Bures-sur-Yvette, France. Received: 6 February 998 / Accepted: 4 April 998 Abstract: The Berry Tabor conjecture asserts that local statistical measures of the eigenvalues j of a generic integrable quantum system coincide with those of a Poisson process. We prove that, in the case of a rectangle billiard with random ratio of sides, the sum N / j N exp(πi jτ) behaves for τ random and N large like a random walk in the complex plane with a non-gaussian it distribution. The expectation value of the distribution is zero; its variance, which is essentially the average pair correlation function, is one, in accordance with the Berry Tabor conjecture, but all higher moments ( 4) diverge. The proof of the existence of the it distribution uses the mixing property of a dynamical system defined on a product of hyperbolic surfaces. The Berry Tabor conjecture and the existence of the it distribution for a fixed generic rectangle are related to an equidistribution conjecture for long horocycles on this product space. Contents Introduction...7. Billiards in a rectangle The main results Basic definitions and notations...77 Invariance Properties of the Form Factor Regime I: τ / ɛ Regime II: / ɛ τ /+ɛ Ergodicity, Mixing and Equidistribution Regime III: τ const The expectation value...88 Supported by the European Post-Doctoral Institute for the Mathematical Sciences, the Engineering and Physical Sciences Research Council (Grant #GR K995), the European Commission (TMR Marie Curie Grant), and the Basic Research Institute in the Mathematical Sciences, Hewlett-Packard Laboratories, Bristol. Permanent address: School of Mathematics, University of Bristol, Bristol BS8 TW, UK. ( J.Marklof@Bristol.ac.uk)

2 7 J. Marklof 6. The it distribution smooth cut-off functions The it distribution general cut-off functions Random walks Rational α The expectation value The it distribution.... Introduction Let 3... be a sequence of numbers satisfying #{j : j },, () which means that the average spacing between adjacent levels is asymptotically unity. One quantity measuring the randomness of the deterministic sequence { j } j is the consecutive level spacing distribution, which is defined by P (s, N) = N N δ(s j+ + j ), () j= where δ(x) is the Dirac mass. The it distribution of P (s, N) for N (if it exists) shall be denoted by P (s). That is, for any sufficiently nice test function h, N P (s, N) h(s) ds = P (s) h(s) ds. (3) Berry and Tabor conjectured [5] that, when the sequence { j } j is a sequence of eigenvalues of a quantum Hamiltonian, whose classical dynamics is integrable, then the it distribution P (s) should in general coincide with the one for a random sequence generated by a Poisson process, i.e., P Poisson (s) =exp( s). This is particularly interesting, because the it distribution for systems, which are not integrable but chaotic, is expected to be the Gaudin distribution for the eigenvalues of random matrices [8], which is approximately described by Wigner s surmise. For the GUE ensemble, say, it reads P GUE (s) 3 π s e 4 π s. Obvious examples for non-generic integrable systems, which violate the Berry Tabor conjecture, are two-dimensional harmonic oscillators, as was already noted by Berry and Tabor [5] and later studied in more detail by Pandey et al. [49], Bleher [7, 8] and Greenman [3]. Further negative examples are Zoll surfaces, where, like on the sphere, all geodesics are closed and have the same length. In the case of the sphere the eigenvalues of the (negative) Laplacian are (after rescaling) E l,m = l(l + ), m = l,...,l, hence The spacings of two-dimensional harmonic oscillators are directly related to the spacings between the fractional parts of the sequence nθ, which had been studied earlier, see [6] for a survey.

3 Spectral form factors of rectangle billiards 7 with multiplicity l +. Label these numbers in increasing order by,,...; due to the high multiplicity one thus has P (s) =δ(s). (4) The same result holds for all other Zoll surfaces where the eigenvalues are extremely clustered around the values l(l + ), compare Duistermaat and Guillemin [6], Weinstein [65], Colin de Verdière [3]. Results in favor of the Berry Tabor conjecture are rare and can so far be only proved for the pair correlation density 3, R (s, N) = N N δ(s j + k ), (5) j,k= which measures the spacings between all elements of the sequence and is therefore not a probability distribution. In the case of a random sequence from a Poisson process, R (s, N) converges to the iting density 4 R Poisson (s) =δ(s)+, which is consequently the expectation for integrable systems. In other words, N R (s, N) h(s) ds = h() + h(s) ds, (6) for a suitable class of test functions h. This variant of the Berry Tabor conjecture with respect to R (s) was verified by Sarnak [54] for the eigenvalues of the Laplacian on almost every flat torus (almost every with respect to Lebesgue measure in the moduli space of two-dimensional flat tori), but he simultaneously disproved the conjecture for a set of second Baire category 5. His result was recently extended to four-dimensional tori by VanderKam [63, 64]. Similar studies in this direction are due to Rudnick and Sarnak [5], whose results can be related to the eigenvalues of boxed oscillators, and Zelditch [66], who considers the level spacings for quantum maps in genus zero. It should be pointed out that, although the above results hold almost everywhere in the corresponding parameter spaces, the Berry Tabor conjecture could not be proved for a specific example. 6 For a more detailed up-to-date review on these topics see Sarnak s lectures [55, 56]. The Berry Tabor conjecture can only be expected to hold for local statistics such as P (s)orr (s), i.e. statistics which only measure the independence of eigenvalues on The local correlations of the eigenvalues of each individual cluster are studied by Uribe and Zelditch [6]. 3 Sinai [59] and Major [45] showed that the statistics of lattice points in certain generic strips follow Poisson statistics in all moments. The boundary of these domains is not twice differentiable and looks like a trajectory for Brownian motion. Spectra of integrable systems like integrable geodesic flows are, however, related via EBK quantization to lattice points in domains with piecewise smooth boundary []. 4 The delta mass δ(s) is a result of our definition which counts spacings between equal elements whose spacing is trivially zero. The interesting part is the. 5 Sets of second Baire category are all sets which are not of first Baire category, and the latter are sets, which are countable unions of nowhere dense sets, so pretty sparse in the topological sense. 6 During the completion of this manuscript I have learned from A. Eskin that it is possible to prove relation (6) for rectangle billiards with ratio of sides α / and α diophantine (e.g. α = ), see [3]. This remarkable result is consistent with Conjecture. (Sect. ).

4 7 J. Marklof the scale of the mean level spacing (which by virtue of () is unity and thus independent of N). Non-local statistics like the number variance (L) are well known to violate the Poisson prediction due to non-universal long-range correlations, see Berry [6] and Bleher and Lebowitz [6, 7]. A different type of non-local statistics is connected with the fluctuations of the energy-level counting function ( the spectral staircase ) around its mean value. It was shown that for certain integrable systems 7 these fluctuations are non-gaussian, even though the local statistics follow in general the Poisson prediction. 8 Fig.. A walk of N = 6 unit steps in the complex plane In the present article, we shall study similar non-local statistical properties, which are, however much closer linked to the local level spacing statistics. The central object of our investigation will be the spectral form factor K (τ,n) (also called pair correlation form factor), which is defined as the Fourier transform of the pair correlation density, with e(z) e πi z, hence, The sum K (τ,n)= R (s, N) = K (τ,n)= N / R (s, N) e(τ s)ds, K (τ,n) e( sτ)dτ N j= e( j τ). (7) N e( j τ) (8) j= may be viewed as a walk in the complex plane (Fig. ), consisting of N steps of unit length, whose direction is determined by the phases ξ j =π j τ. In the case when the j come from a Poisson process, the probability of finding the end point after N steps outside a disk of radius NR (R is a fixed constant) has by virtue of the classical 7 E.g. billiards in a rectangle and tori [34, 9,, 4, 3], Liouville surfaces [4, 5], other surfaces of revolution [], and Zoll surfaces [57]. For a survey see []. 8 In contrast, for chaotic systems the fluctuations of the spectral staircase are conjectured to be Gaussian [4, 3], based mainly on numerical evidence.

5 Spectral form factors of rectangle billiards 73 central it theorem in the it N agaussian it distribution in the complex plane. This means in particular that N ProbPoisson {K (τ,n) >R}= R e r dr =e R. (9) If, however, the j are given by a deterministic sequence, we can test the independence of the j by considering the distribution of the sum (8), i.e. the distribution of endpoints of the corresponding walk, for different values of τ. That is, we throw τ at random with probability density ρ, and ask if, as above, N Probρ {K (,N)>R}=e R. () Clearly, the answer to this question may now also depend on correlations between the j, which do not only appear on the scale of the mean level spacing but as well on scales in units of N γ, for some power γ, say. That is the reason why we have classified this statistic as non-local. For τ random as above, relation (6) can be reformulated as N Eρ K (,N)=+ρ(), () where E ρ denotes the expectation, and h =ˆρand ρ are related by Fourier transformation, ˆρ(s) = ρ(τ) e(sτ)ds. The statistical properties of the form factor K (τ,n) have received great attention in the quantum chaos literature, see e.g. [43,, 44, 7, 5, ] and references therein. It is generally believed that the normalized fluctuations of the form factor for a chaotic system are of Gaussian nature [5, 4], cf. also footnote 8 in []. The situation for integrable systems seems to be more subtle. We shall see that, even though a generic rectangle billiard is likely to follow Poisson statistics locally [5,, 7], the fluctuations of the form factor are not Gaussian but have a it distribution with algebraic-logarithmic tail. 9 On the other hand, numerical studies of the circle billiard exhibit Gaussian fluctuations of the form factor [6]. If these are truly Gaussian for the circle or other generic integrable cases (e.g. Liouville surfaces), remains an interesting open problem... Billiards in a rectangle. The quantum energies of a rectangle billiard are given by the eigenvalues of the negative Laplacian = q q with Dirichlet conditions on the boundary of the rectangle. The eigenvalues scale trivially with the area of the rectangle, so the only parameter which will enter the theory is the side ratio α /,<α. (The reason why we work with α rather than the side ratio itself will become apparent later.) 9 Similar deviations from a Gaussian distribution have been observed by Casati et al. [], who calculated (numerically) the fluctuations of the difference P (s, N) P Poisson (s). Other non-local statistics for rectangle billiards, such as the number variance and the fluctuations of the spectral staircase have been considered e.g. by Casati et al. [, ], Berry [6], Bleher et al. [, 4, 3, 6, 7].

6 74 J. Marklof Taking the area to be 4π we have eigenvalues E (α) m,n = π 4 (α/ m + α / n ), m,n N, which we label in increasing order and with multiplicity by < (α) < (α) (α) 3... This sequence clearly satisfies (), since the number of lattice points in an ellipse of area is asymptotically. More precisely, we have #{j : (α) j } = π / (α /4 + α /4 ) / + O γ ( γ ), () where the relation is conjectured to hold for any γ> 3 4 ; the best bound so far, γ> 73, is due to Huxley [36]... The main results. Let us view τ as a random variable, which is distributed on the compact interval I R with a piecewise continuous probability density ρ, and α as a random variable with piecewise continuous probability density σ on the compact interval A R + {}. The expectation E ρ,σ for the random variable f(τ,α) is then defined as E ρ,σ f = f(τ,α)ρ(τ) σ(α)dτ dα, I and its probability to be greater than R by Prob ρ,σ {f>r}= D R (τ,α)ρ(τ) σ(α)dτ dα, with the distribution function A D R (τ,α)= I A { if f(τ,α) >R otherwise. Theorem.. There exists a decreasing function : R + R + with ()=, (R)dR =, (R) cr log R (for R ), discontinuous only for at most countably many R, such that, N Probρ,σ {K (,N)>R}= (R), except possibly at the discontinuities of (R). The constant c is given by the expression c = π R 6 e(t u + t u ) d t 4 d u. (3) t +t

7 Spectral form factors of rectangle billiards 75 Remarks. (A) The condition (R) dr = is consistent with a Gaussian it distribution and thus consistent with the fact that the pair correlation function is the one for Poisson random numbers. (B) The condition (R) cr log R implies that the it distribution is non- Gaussian. In particular, the higher moments R k d (R) (k>) diverge. This divergence has the following origin. Just as the moment k = is related to the pair correlation function R (s, N) (i.e. counting spacings), the moment k = is related to the density N N j,j,k,k = and thus to (with the above test function h) N N j,j,k,k = δ(s j j + k + k ), (4) h( j + j k k ), (5) which amounts to counting the number of quadruples of eigenvalues such that j + j k k is in the interval [ a, a], say. In the case of a rectangle billiard, this number grows like (N log N), due to number-theoretic degeneracies: recall that in this case we count essentially the integers m,...,m 4,n,...,n 4 N with (m + αn )+(m +αn ) (m 3 + αn 3) (m 4 + αn 4) (6) in some interval. The number of solutions of m + m = m 3 + m 4, n +n = n 3 +n 4 with m,...,m 4,n,...,n 4 N grows like (N log N), due to Landau s classical result on the number of ways of writing an integer as a sum of two squares. Hence the number of solutions of (6) in any arbitrarily small interval [ ɛ, ɛ] grows like (N log N), hence the divergence of the second moment, at least for test functions h with h(). But since h is the Fourier transform of a probability density, we in fact have h() =. This explains the divergence of the moments k = and higher. Consequently, we cannot employ the method of moments to prove the above theorem. We shall instead use another approach based on the transformation formulas of theta functions, relating the existence of the it distribution to the mixing property of certain flows, see Sect. 5 for details. This is essentially the same idea as in the proofs of the it theorems of theta sums N e(n x) n= in [46, 47, 48], which is now generalized to theta sums with more variables. The methods presented here could be further generalized to Siegel theta sums of arbitrary quadratic forms Q(ξ) ofdvariables, e(q(ξ) τ). ξ Z d N

8 76 J. Marklof Here, N denotes some suitable domain R d, which is magnified by a factor of N. It would be interesting to see, in which cases the values of the above theta sums have a it distribution for τ random. The expectation value of this it distribution is related to the quantitative version of the Oppenheim conjecture for Q(ξ), cf. Eskin et al. [9, 3] and Borel s survey [9]. (C) Theorem. implies that there cannot be a set of α of non-zero Lebesgue measure, for which (for fixed α and random τ) Prob ρ {K (,N) >R}converges to e R.We instead have good reasons to believe the following to be true for those α, which are diophantine, i.e., badly approximable by rationals. Conjecture.. Theorem. even holds when α is not random but fixed, as long as α is diophantine. That means in particular N Probρ {K (α) (,N)>R}= (R), with the same function (R) as before. The truth of this conjecture is related to the equidistribution of horocycles in the product space Ɣ\ PSL(, R) Ɣ\ PSL(, R) (for details see Sect. 5, in particular Conjecture 5.6, and Sect. 6). The conjecture is obviously false for rational α = p q. The following theorem follows from the results in [39, 46], see Sect. 7 for details. Theorem.3. Let ψ be piecewise continuous and of compact support. Then there exists a decreasing function ( p q ) : R + R + with ( p q ) ()=, (p q ) (R) c (p q ) R (for R ), discontinuous for at most countably many R, such that, p q ) Probρ {K ( (,)>R}= (p q ) (R), except possibly at the discontinuities of ( p q ) (R). In the special case α = p q = the constant c( p q ) reads c () =/π. Remark. The fact that now (R) dr = diverges is due to the well known degeneracy of values of rational quadratic forms at integers (cf. previous remark B). In particular, one has for the pair correlation function (Proposition 7.) E ρ K ( p q ) (,N) b (p q ) log N (N ). (7) For α = p q = we have b() =/π. The logarithmic divergence resembles the average number of ways to write an integer as a sum of two squares, which is a classical result by Landau, cf. the previous Remark (B). As a consequence, the consecutive level spacing distribution is P (s) = δ(s) for all rational α. For more information on two-level statistics for the square billiard, cf. also Connors and Keating [4]. The precise definition of a diophantine number will be given later (Sect. 5). The set of diophantine numbers is of full Lebesgue measure in R.

9 Spectral form factors of rectangle billiards Basic definitions and notations. The expressions x a y and x = O a (y) both mean there exists a constant C a (which may depend on some additional parameter a) such that x C a y. The notation x = O(y ) is an abbreviation for x = O M (y M ) for every M M, for some suitably large constant M.Apiecewise continuous function is discontinuous only on a set of measure zero, and bounded on all compact sets. We denote by S(R n ) the Schwartz class on R n, i.e. the space of smooth functions f(t,...,t n ) which decrease rapidly when t +...+t n. The same must hold for all derivatives of f.. Invariance Properties of the Form Factor The pair correlation density R (s, ) and the form factor K (τ,) measure the statistics of levels in an energy window [,]. For technical reasons it is convenient to smooth this window, i.e., choose a smooth cut-off function ψ C (R + ), which is rapidly decreasing at and consider the smoothed (in ) pair correlation density R,ψ (s, ) = and the smoothed form factor j, k ψ( j ) ψ( k ) δ(s j + k ), (8) K,ψ (τ,)= / j ψ( j )e( jτ). (9) In the case of the rectangle billiard the form factor has the explicit expression K (α),ψ (τ,)= / m,n N π 4 ψ( (α/ m + α / n ) ) For symmetry reasons, we can write the sum as 4 = m,n N m,n Z m=,n Z e( π 4 (α/ m + α / n ) τ). () m Z,n= + m=,n= The form factor can now be expressed in terms of the theta functions f (z) =y /4 n Zf(ny / )e(n x), (). and f (z ; z )=y /4 y /4 m,n Z f(my /,ny / )e(m x +n x ), () where z j = x j +iy j is a complex variable. Setting and f(t) =ψ(t ), f(t,t )=ψ(t +t ),

10 78 J. Marklof we have z = π 4 α/ (τ +i ), z = π 4 α / (τ +i ), K (α),ψ (τ,)= 6 (π 4 ) / f (z,z ) /4[ ( π 4 α/ ) /4 f (z )+( π 4 α / ) /4 f (z ) ] + / ψ(). (3) It is intuitively clear that in the it the most important contributions should come from the two-variable theta sum. However, for special values of α and τ this need not be the case, but the set of exceptions is fortunately of measure zero. The following lemma follows from standard estimates on theta sums [33]. Lemma.. For almost all α and τ (with respect to Lebesgue measure) and any ɛ> we have K (α),ψ (τ,)= ˆK (α),ψ (τ,)+o α,τ,ɛ( /4+ɛ ), with ˆK (α),ψ (τ,)= 4π f(z,z ). The transformation formulas of theta functions () in one complex variable were the starting point of the studies in [46, 47, 48], and can be readily generalized to the two-variable sum f (z ; z ), by considering each variable separately. Before stating the transformation formulas, we have to introduce some geometry. Every element g in the Lie group PSL(, R) =SL(,R)/{±} has a unique Iwasawa decomposition ( )( )( ) x y / cos φ sin φ g = y /, sin φ cos φ where z = x +iyis a point in the upper half plane H = {z = x +iy:x, y R, y>}, and φ [,π) parametrizes the circle S, so the underlying manifold of PSL(, R) may be identified with the manifold H S. The invariant volume element (Haar measure) reads in this choice of coordinates dµ(g) = dx dy dφ y. By virtue of the relation ( )( )( ab x y / )( ) cos φ sin φ cd y / sin φ cos φ ( ) ( x y = / y / ) (cos ) φ sin φ sin φ cos φ, (4) where x +iy = a(x+iy)+b c(x+iy)+d, φ = φ+arg[c(x+iy)+d],

11 Spectral form factors of rectangle billiards 79 the action of PSL(, R)onH S is canonically given by g (z,φ) = ( gz, φ +arg(cz + d) ), g = where g acts on z H by fractional linear transformations, i.e., gz= az + b cz + d. The theta group Ɣ θ, which is generated by the elements ( ) and ( / ), ( ) ab, (5) cd corresponding to the transformations ( ) (z,φ) = ( z+,φ), ( ) / (z,φ) = ( 4z,φ+argz), is an example of a discrete subgroup of PSL(, R), such that the quotient manifold M θ = Ɣ θ \ PSL(, R) ={Ɣ θ h : h PSL(, R)} has finite volume µ(m θ )=π, but is not compact (with respect to the measure introduced above). A fundamental region of Ɣ θ in H is (Fig. ) F θ = {z H : x < /, z > /}. (6) There are two cusps which are represented by the points at and ( is equivalent to ). It is well known that every rational point on the boundary Im z = of H is Ɣ θ -equivalent to one of the cusp points, i.e. there is always an element ( ) ab Ɣ cd θ y -/ / Fig.. The fundamental region F θ = {z H : x < /, z > /} in H for the theta group Ɣ θ. The cusps are represented by the points at and. The thin lines indicate the symmetries of the surface x

12 8 J. Marklof such that a p q + b c p q + d = or. (7) We first review some properties of one-variable theta sums, cf. [46, 47, 48]. For f S(R), define the theta function f as a function on H [, 4π)by f (z,φ) =y /4 n Zf φ (ny / )e(n x), (8) where f φ (t) = G φ (t, t ) f(t ) dt, (9) R with the harmonic oscillator Green function [ ] G φ (t, t )= / e( σ φ /8) sin φ / (t + t ) cos φ tt e, sin φ where σ φ =k+ when kπ<φ<(k+)πwith k Z. Proposition.. Let f S(R). Then f (z,φ) is infinitely differentiable and satisfies the following functional relations: f (z +,φ)= f (z,φ), f ( 4z,φ+argz)=e iπ/4 f (z,φ). The function f may thus be viewed as an infinitely differentiable function on the manifold M θ = Ɣ θ \ PSL(, R). The second relation implies f (z,φ + π) = i f (z,φ). Since our theta function is smooth, it is bounded except in the cusps. In order to state the asymptotic behaviour of the function in the cusp at, we introduce a set of new coordinates, (w, θ) =( (4z ),φ+arg(z /)), in which the cusp at is represented as a cusp at infinity. That is, v =Imwis the coordinate pointing into the cusp, and u =Rewthe one orthogonal to it. Proposition.3. Let f S(R). Then { y /4 f φ () + O(y ) (y> f (z,φ) = O(v ) (v> ) ).

13 Spectral form factors of rectangle billiards 8 The ranges for which the above relations hold, cover the entire fundamental region {z H : <x<, z >/, z >/}. (This new region is obtained from the old one F θ by shifting the left half by x x +.) Within these ranges, the relations are uniform in (z,φ) (st rel.), (w, θ) (d rel.). For f S(R ), define the theta function f as a function on H [, 4π) H [, 4π) by f (z,φ ;z,φ )=y /4 y /4 m,n Z f φ,φ (my /,ny / )e(m x +n x ), (3) where f φ,φ (t,t )= G φ (t,t )G φ (t,t )f(t,t )dt dt, R (3) with the same harmonic oscillator Green function G φ (t, t ) as before. It can be readily verified that f φ,φ (t,t )isagain S(R ) (use partial integration in t and t ), so the sum defining f is rapidly convergent for every (φ,φ ). Proposition.4. Let f S(R ). Then f (z,φ ;z,φ )is infinitely differentiable and satisfies the following functional relations: f (z +,φ ;z,φ )= f (z,φ ;z,φ ), f ( 4z,φ +argz ;z,φ )=e iπ/4 f (z,φ ;z,φ ), f (z,φ ;z,φ )= f (z,φ ;z,φ ). The function f may thus be viewed as an infinitely differentiable function on the manifold M θ = Ɣ θ \ PSL(, R) Ɣ θ \ PSL(, R). The second relation implies f (z,φ +π;z,φ )= i f (z,φ ;z,φ ). As mentioned above, the proof of the proposition follows exactly the lines of the analogous proposition for one-variable theta functions, compare e.g. [47]. As a fundamental region of Ɣ θ Ɣ θ we choose the set The four cusps of codimension two are represented by F = F θ [,π) F θ [,π). (3) (,φ ;z,φ ), (,φ ;z,φ ), (z,φ ;,φ ), (z,φ ;,φ ); every point of the form ( p q,φ ;z,φ ), (z,φ ; p q,φ ), p q Q is equivalent to one of those four, compare (7).

14 8 J. Marklof Proposition.5. Let f S(R ). Then f (z,φ ;z,φ )= (y y ) /4 f φ,φ (, ) + O((y y ) ) (y >,y > ) O((v v ) ) (v > =,v > ) O(y /4 v ) (y >,v > ) O(v y /4 ) (v >,y > ). These relations are uniform in (z,φ ;z,φ ) (st rel.), (w,θ ;w,θ ) (d rel.), (z,φ ;w,θ ) (3d rel.), (w,θ ;z,φ ) (4th rel.). The proof follows directly from Proposition in [46]. The inner products and norms of the L spaces in question are defined as usual by (f,g) L (R ) = f gd x, f L (R ) = (f,f) L (R ) (33) R and (F, G) L (M θ ) = F Gd µ, F L (M θ ) = (F, F) L (M θ ). (34) M θ By virtue of the last proposition, it is clear that f is in L (M θ ). If f is even, i.e. f(t,t )=f( t,t )=f(t, t ), we have the relation π f L (M θ ) = f L (R ), (35) compare [46], Proposition Regime I: τ / ɛ We begin with the regime where τ C I γ with γ>, and C I an arbitrary constant. The behaviour of the form factor is entirely determined by the values of the theta function in the cusp at, since ˆK (α),ψ (τ,)= 4π f(z,; z, ) = 4π f (, arg z ;, arg z ) 4z 4z with = 4z πα / = 4z πα / τ +i τ +, arg z =arg(τ+i ), τ +i τ +, arg z =arg(τ+i ). For, the imaginary part of /4z j becomes infinitely large, so Proposition.5 is applicable, yielding ˆK (α),ψ (τ,)= 4π f arg(τ+i ),arg(τ+i )(, ) τ + + O( ) (36) which holds uniformly for τ C I γ. It is easy to see that

15 Spectral form factors of rectangle billiards 83 f arg(τ+i ),arg(τ+i )(, ) =4 (τ + ) e[(t + t )τ]f(t,t )dt dt Therefore ˆK (α) =4 (τ + ) π e(rτ) ψ(r) dr. (37),ψ (τ,)= e(rτ) ψ(r) dr + O( ) (38) for uniformly in τ C I γ. In this regime, the form factor looks therefore asymptotically as a delta mass at zero: upon applying a test function ρ we have CI γ γ CI ρ(τ) ˆK (α),ψ (τ,)dτ ρ(τ/) e(rτ) ψ(r) dr dτ, C I γ C I γ and with ρ(τ/) ρ() for τ [ C I γ,c I γ ] the above reduces to ρ() by Parseval s equality. In summary, CI γ e(rτ) ψ(r) dr dτ = ρ() γ CI ρ(τ)k (α),ψ (τ,)dτ ρ(τ) ˆK (α),ψ (τ,)dτ C I γ C I γ ρ() ψ(r) dr ψ(r) dr. (39) Regime I is usually referred to as the saturation regime, since the number variance (L) saturates in this regime, in contrast to the prediction for a Poisson process [6, 6]. 4. Regime II: / ɛ τ /+ɛ Regime II is defined as the region where C I / ɛ τ C II /+ɛ, with C I and C II arbitrary positive constants. From the discussion of Regime I it is clear that the form factor in Regime II can at most grow like K,ψ (τ,)=o ɛ ( ɛ ), (4) which is obtained from (38) for τ = / ɛ. The other τ-values in Regime II are bounded farther away from the cusp, so that the above indeed gives an upper bound. The most interesting part of Regime II is when C I / τ C II /. Let us put ω = τ /. Then we have for large, and = 4z j πα ±/ ω / +i ω + πα ±/ ( ω / +iω )

16 84 J. Marklof arg z j =arg(ω / +i ) +. Since the series defining f (z,φ ;z,φ ) converges uniformly for y and y in compacta, we finally obtain K (α),ψ (τ,) ˆK (α),ψ (τ,) ˆK (/α),ψ ( 4 / π ω, 4 π ) ( ) (4) ω uniformly for ω = τ / [C I,C II ]. Before turning to Regime III (τ const.), where we shall study the fluctuations of the form factor around its mean, we will have to prove some facts about the equidistribution of certain sets in M θ. 5. Ergodicity, Mixing and Equidistribution In this section we discuss some ergodic properties of the geodesic flow on hyperbolic surfaces of finite area, whose unit tangent bundle is represented by the quotient M = Ɣ\ PSL(, R), where Ɣ is a discrete subgroup, such as the theta group Ɣ θ. The surface should only have a finite number of cusps ( ), and we assume that, if there is at least one cusp, the half-plane coordinates are chosen in such a way that one of the cusps appears as the standard cusp of unit width at. Consider the following three flows on M = Ɣ\ PSL(, R), defined by right translation, ( ) g g t, t e t/ = e t/, (4) and g g t ±, t + = ( ) t, t = ( ). (43) t These flows actually represent the geodesic and the (positive and negative) horocycle flow on the unit tangent bundle of the surface Ɣ\H, which can be identified with Ɣ\ PSL(, R) (the angle φ is in fact times the orientation angle of the tangent vector). These flows are well known to be ergodic and mixing [5]. The mixing property can be stated as follows. Proposition 5.. Let F, G L (M). Then F (g) G(g t ) dµ(g) = Fdµ Gdµ. t ± M µ(m) M M The mixing property has an interesting consequence for the asymptotic distribution of long arcs of horocycles, which is stated in the following corollary. In fact, the investigation of measures concentrated along unstable fibers (which in our case are the horocycles) is a central issue in the theory of dynamical systems. Our proof will follow an idea of Eskin and McMullen [8], Theorem 7..; for related methods, cf. Kleinbock and Margulis [4], Sect.... and references therein.

17 Spectral form factors of rectangle billiards 85 Corollary 5.. Let F be bounded and piecewise continuous on M, and h be piecewise continuous and of compact support on R. Then, for any g M, h(u) F (g u + t ) du = Fdµ h du. t µ(m) R Proof. Every element g = ( ab cd) SL(, R) with d can be written as a product M g = ± u + a b. (44) Let F be continuous on PSL(, R), left-invariant under Ɣ and compactly supported when viewed as a function on M = Ɣ\ PSL(, R). Let furthermore H be the function on PSL(, R) defined for some fixed ɛ>by { h(u) H(g)= ɛ χ(a ɛ )χ(b) for g = u + a b for g = ( ab c), where χ is the characteristic function of the interval [, ]. Consider the integral H(g) F (g g t ) dµ(g) = H( u + a b ) F (g u + a b t ) dµ. PSL(,R) PSL(,R) (45) The bi-invariant Haar measure in these coordinates is given by dµ =e a da db du. (46) R Step A. Now using the relation b t = t b et the integral transforms to H( u + a b ) F (g u + a+t b et ) dµ. (47) PSL(,R) The distance (with respect to the invariant metric on PSL(, R)) between the points g u + a+t b et and g u + a+t is b e t, hence F (g u + a+t b et )=F(g u + a+t )+O(be t ), (48) where the implied constant does not depend on u, a, b or t, for F is uniformly continuous. For t large we thus have H( u + a b ) F (g u + a+t b et ) dµ PSL(,R) = h(u) R ɛ χ(a ɛ ) F (g u + a+t )e a da du + O(e t ). (49) Step B. We can rewrite (45) as H(g g) F (g t ) dµ(g) = PSL(,R) M ( γ Ɣ H(g γg) ) F(g t )dµ(g) (5)

18 86 J. Marklof since F and dµ are Ɣ-left-invariant. The latter, by the mixing property, has the it (notice that the function G(g) = γ Ɣ H(g γg)isɣ-left-invariant) Fdµ µ(m) M Mγ Ɣ = H(g γg) dµ(g) Fdµ µ(m) M = µ(m) M Hdµ PSL(,R) Fdµ h(u) R ɛ χ(a ɛ )e a da du. (5) Step C. Combining Steps A and B we conclude that h(u) χ(a) F (g u + ɛa+t )e ɛa da du t R = Fdµ h(u)χ(a)e ɛa da du. (5) µ(m) M R By the uniform continuity of F, given any δ>, we find an ɛ>such that h(u) χ(a) F (g u + ɛa+t )e ɛa da du δ R < h(u) χ(a) F (g u + t ) da du R < h(u) χ(a) F (g u + ɛa+t )e ɛa da du + δ. (53) R Since the its t on the left and right hand side exist for every fixed ɛ, and differ by δ, which can be made arbitrarily small, the it h(u) F (g u + t ) du = Fdµ h(u)du (54) t R µ(m) M R must exist as well. In order to relax the condition of compact support for F notice that the assertion trivially holds for F const. One can then again use the inclusion principle to relax both the compact support hypothesis and the continuity hypothesis. Remarks. (A) It is crucial that the it is t. The it t + diverges, since in this case the support of the measure converges towards a single it point on the boundary of H. (B) For g = the assertion of the corollary can be stated in the (z,φ)-coordinates as y R h(x)f (z,) dx = µ(m) M Fdµ R hdx. (55) In the case when the average is taken over a closed horocycle and h, this was proved by Sarnak [53]. Hejhal [35] extended the result to h of the above class, but still for closed horocycles and for F which are independent of φ, i.e. functions on Ɣ\H. His proof involves estimates on Poincaré series of weight zero.

19 Spectral form factors of rectangle billiards 87 The observation of the previous corollary shall now be extended from averages over translates of horocycles to averages over translates of more general curves, which are given by the equation (in suitable coordinates) y =e f(x), where f is a bounded real function. The case f const corresponds to horocycles. Corollary 5.3. Let F, h be as before, and let f : R R be continuous on the support of h. Then, for any g M, h(u) F (g u + t+f(u) ) du = Fdµ h du. t R µ(m) M R That is, for g =, h(x)f (x +iye f(x),) dx = y R Fdµ hdx. µ(m) M R Proof. Repeat the proof of the previous corollary with { h(u) H(g) = ɛ χ(a f(u) ɛ )χ(b) for g = u + a b for g = ( ab c), which still has compact support, since f is bounded on the support of h. The only main difference to the previous proof is in Step C where we have instead h(u) χ( a f(u) ) F (g u + a+t ) da du ɛ R ɛ = h(u) χ(a) F (g u + ɛa+t+f(u) ) da du, (56) R and the conclusion is the same as before, for f is bounded on the domain of integration. The next step is to define flows on the product space by the diagonal action M = Ɣ\ PSL(, R) Ɣ\ PSL(, R) Ɣ\ PSL(, R) Ɣ\ PSL(, R) Ɣ\ PSL(, R) Ɣ\ PSL(, R), (57) (g,g ) (g,g ) t := (g t,g t ) and (g,g ) (g,g ) t ± := (g t ±,g t ±). We shall still call these flows geodesic or horocyclic, respectively. Each one is the direct product of two mixing dynamical systems, and thus mixing itself [5]: Proposition 5.4. Let F, G L (M ). Then t ± M F (g,g )G((g,g ) t )d µ(g,g )= µ(m ) M Fd µ M Gd µ.

20 88 J. Marklof The analogue of Corollaries 5. and 5.5 is the following: Corollary 5.5. Let F be bounded and piecewise continuous on M, h be piecewise continuous and of compact support on R, and f =(f,f ):R R continuous on the support of h. Then, for any (g,g ) M, h(u,u )F(g u + t+f(u,u),g u + t+f(u,u) ) d u t R = µ(m Fd µ hd u. ) M R The proof of Corollary 5. obviously generalizes to this case. For (g,g )=we have y h(x,x )F(x +iye f(x,x), ; x +iye f(x,x), ) d x R = µ(m Fd µ hd x. (58) ) M R There are reasons to believe that the above equidistribution results do not only hold for two-dimensional averages over (x,x ), but even for averages over one-dimensional lines given by x = ηx. In the case when the lattice group Ɣ is a congruence subgroup of PSL(, R) (such as the theta group Ɣ θ ), it has to be assumed, however, that η is badly approximable by rationals, i.e. η p q C q κ for all rationals p q and some κ. Numbers η satisfying this condition are called diophantine. The reason for the necessity of this condition will become clearer in Sect. 7. Conjecture 5.6. Suppose M = Ɣ\ PSL(, R) with Ɣ a congruence subgroup of PSL(, R). Let F be bounded and piecewise continuous on M, h be piecewise continuous and of compact support on R.Ifηis diophantine, we have for all y,y >, h(x)f (x +iy y, ; ηx +iy y, ) dx = y R µ(m Fd µ hdx. ) M R The above observations on the equidistribution of certain sets will now be applied to understand the value distribution of the form factor in the most subtle Regime III, where the order of magnitude of τ is independent of. 6. Regime III: τ const 6.. The expectation value. The following proposition is the analogue of Sarnak s Proposition.(A) on the weak convergence of the pair correlation function in the case of a two-dimensional family of tori [54]. Here, the weak convergence is with respect to our one-dimensional family of rectangles, parametrized by α. (Recall the definition of ρ and σ in Sect...)

21 Spectral form factors of rectangle billiards 89 Proposition 6.. Let ψ be piecewise continuous and of compact support. Assume furthermore ρ is continuous at. Then Eρ,σ K,ψ (,)= ( +ρ() ) ψ(r) dr. Proof. We have to estimate the integral π / 4 ψ( (α/ m + α / n ) ) m,n N e( π 4 (α/ m + α / n ) τ) ρ(τ) σ(α) dτ dα. (59) Let us first assume all functions involved are infinitely differentiable. For later purposes it is furthermore convenient to assume that the functions ρ and σ are not necessarily probability densities, i.e., have averages also different from one. This freedom will be needed when we approximate piecewise continuous probability densities from above/below by smooth ρ and σ. To be able to use the inclusion principle for ψ, we have to replace (59) by the slightly more general expression m,n,m,n N π ψ ( C I / ɛ 4 (α/ m + α / n ) π ) ψ ( 4 (α/ m + α / n ) e ( π 4 (α/ (m m )+α / (n n )) τ ) ρ(τ) σ(α) dτ dα. (6) We get back to (59) by setting ψ = ψ = ψ. In the sequel, ψ and ψ are taken to be C with compact support. Let us split the domain of integration over τ into the three parts corresponding to Regime I, Regime II and Regime III, that is, CI / ɛ CII /+ɛ = + + C I / ɛ C II /+ɛ and similarly for the integral over the negative axis. The first integral is similar to the one calculated in Sect. 3 (where ψ = ψ = ψ), see (39), and we have for large, CI / ɛ ρ() ψ (r) ψ (r)dr σ(α) dα. (6) This result can be obtained most easily by replacing the modulus squared of the theta function, f (z,φ ;z,φ ), by the product ) f (z,φ ;z,φ ) f (z,φ ;z,φ ), (6) with f ν (t,t )=ψ ν (t +t ). The function in (6) can still be viewed as a function on M θ, cf. Proposition.4. The second integral vanishes to leading order, due to the bound analogous to bound (4) valid in Regime II (Sect. 4), so CI / ɛ CII /+ɛ + C II /+ɛ C I / ɛ ɛ (C II /+ɛ C I / ɛ ) /+3ɛ. (63)

22 9 J. Marklof We are left with the third integral, where τ C II /+ɛ, which we split into two parts, the diagonal part τ C II /+ɛ m,n N π 4 ψ ( (α/ m + α / n ) ) π 4 ψ ( (α/ m + α / n ) ) ρ(τ) σ(α) dτ dα, (64) which clearly converges for to the desired expression ψ (r) ψ (r) dr ρ(τ) dτ σ(α) dα, and the off-diagonal part, (m,n ) (m,n ) N π ψ ( τ C II /+ɛ 4 (α/ m + α / n ) π ) ψ ( 4 (α/ m + α / n ) e( π 4 (α/ (m m )+α / (n n )) τ) ρ(τ) σ(α) dτ dα. (65) ) Substituting x = π 4 α/ τ, x = π 4 α / τ the last expression equals (m,n ) (m,n ) N ψ ( π 4 π xx C II /+ɛ x x m + x x n ) ψ ( π x x m + x x n ) 4 e(x (m m )+x (n n )) h(x,x )dx dx, (66) where h is related to ρ, σ and the Jacobian of the substitution in the obvious way, h(x,x )= 4 π Let us divide (66) into the sums = (m,n ) (m,n ) N m m,n n N x x ρ( 4 x x )σ( x ). (67) x π x + m m,n =n N + m =m,n n N Partial integration with respect to x and x shows the first sum diverges only logarithmically in, and, after some manipulations, this part of (66) turns out to be bounded by (use partial integration in x,x, and then re-substitute the old variables τ and α) Notice that the sum over m, n converges to the corresponding Riemann integral..

23 Spectral form factors of rectangle billiards 9 m m,n n ɛ τ /+ɛ τ C II /+ɛ ρ(τ) τ ρ(τ) σ(α) dτ dα τ m m n n dτ / ɛ+ɛ τ ρ(τ /+ɛ ) τ dτ / ɛ+ɛ (68) for any ɛ >. The remaining two sums can be shown to be of non-leading order by similar means. We have thus shown so far that for, (m,n ),(m,n ) N π ψ ( 4 (α/ m + α / n ) π ) ψ ( 4 (α/ m + α / n ) ˆρ( π 4 (α/ (m m )+α / (n n ))) σ(α) dα ( ) ψ (r)ψ (r) dr ρ() + ρ(τ) dτ σ(α) dα. (69) Taking finite linear combinations H(r,r ) = ψ j (r )ψ j (r ) of functions of the above type, we have (m,n ),(m,n ) N π 4 H( (α/ m + α / n ), π 4 (α/ m + α / n ) ˆρ( π 4 (α/ (m m )+α / (n n ))) σ(α) dα ( ) H (r, r) dr ρ() + ρ(τ) dτ σ(α) dα. (7) Let us now see how the smoothness condition on ψ can be relaxed to piecewise continuous ψ with compact support. Following the lines of the proof of Theorem 3. in [5], ψ(r )ψ(r ) can now be approximated from above/below by smooth test functions H(r,r ), which are admissible for the previous derivation and for which H(r, r) ψ(r) dr < ɛ. (7) By the inclusion principle the statement of the proposition holds thus for piecewise continuous ψ. The result can be further extended to piecewise continuous ρ and σ by approximating both sides of the relation Eρ,σ K,ψ (,)= ( +ρ() ) ψ(r) dr from above and below using smooth functions ρ ɛ and σ ɛ, which are ɛ-close to ρ and σ, respectively. In the L sense but also in the sense that ρ() ρ ɛ() <ɛ, which is where the continuity of ρ at is required. ) )

24 9 J. Marklof 6.. The it distribution smooth cut-off functions. Let us first consider the simpler case when ψ is a smooth cut-off function. The results will later be extended to more general cut-off functions. We denote by S(R + ) the space of even functions in S(R), restricted to the positive half line R +. Theorem 6.. Let ψ S(R + ). Then there exists a decreasing function : R + R + with ψ ()=, ψ (R)dR = ψ(r) dr, discontinuous only for at most countably many R, such that, Probρ,σ {K,ψ (,)>R}= ψ (R), except possibly at the discontinuities of ψ (R). Theorem 6.3. For R, we have the asymptotic relation ψ (R) =c ψ R log R + d ψ R + O ψ (R ), where c ψ = π R 6 e(t u + t u ) ψ(t + t ) d t 4 d u. R The constant d ψ is given by a more complicated expression, see the proof of this theorem. The proofs of these theorems will be given below. The following conjecture is a direct consequence of the equidistribution hypothesis of Conjecture 5.6, compare the proof of Theorem 6.. Conjecture 6.4. Theorem 6. even holds when α is not random but fixed, as long as α is diophantine. That means in particular Probρ {K (α),ψ (,)>R}= ψ(r), with the same function ψ (R) as before. This conjecture does not hold when α is rational or well approximable by rationals, as we shall see in the next section. Proof of Theorem 6.. Apply Corollary 5.5, i.e. relation (58), where χ D is taken to be the characteristic function of the set D = {(z,φ ;z,φ ) M θ : 4π f(z,φ ;z,φ ) >R}. The boundary of D can only have positive measure for a countable number of R, otherwise this would be a contradiction to the measurability of f. Hence χ D is piecewise continuous except for countably many R. We chose the function h(x,x ) in a way that h(x,x )χ D (x +iye f(x,x), ; x +iye f(x,x), ) d x = D R (τ,α,)ρ(τ)σ(α)dτ dα, (7) where

25 Spectral form factors of rectangle billiards 93 x = π 4 α/ τ, x = π 4 α / τ, y = π 4, x f (x,x ) = log, x x f (x,x ) = log, x and { if K (α),ψ (τ,) >R D R (τ,α,)= otherwise. Since the compact interval A does not contain, the functions f and f are bounded on the domain of integration. Hence, except for at most countably many R, Probρ,σ { ˆK,ψ (,)>R}= ψ (R). (73) To obtain the same result for K,ψ, recall that by virtue of (3) we can express the difference between K,ψ and ˆK,ψ as the sum /4 F (z,φ ;z,φ )+ / F (z,φ ;z,φ ) + 3/4 F 3 (z,φ ;z,φ )+ ψ(), where the F j are sums of products of theta functions and their modulus is therefore majorized by functions on M θ ; it is therefore clear that Probρ,σ { K,ψ (,) ˆK,ψ (,) >ɛ}= (74) for arbitrarily small (but fixed) ɛ>. Hence for any given ɛ>, we have Prob ρ,σ { ˆK,ψ (,)>R+ɛ} Prob ρ,σ {K,ψ (,)>R} Prob ρ,σ { ˆK,ψ (,)>R ɛ}, (75) for every large enough. The its ψ (R+ɛ) and ψ (R ɛ) of the left- and right-hand side exist except for countably many R, ɛ. Since {(z,φ ;z,φ ) M θ : 4π f(z,φ ;z,φ ) =R} has positive measure only for countably many R, the difference ψ (R ɛ) ψ (R + ɛ) can be made arbitrarily small for suitable small ɛ>, except for countably many R. Finally, the relation ψ () = holds by definition, 3 and the integral of ψ can be calculated as follows (notice that µ(m θ )=π4 ): 3 It might happen that ψ (ɛ) C< for arbitrarily small ɛ, but since we allow ψ (R) to be discontinuous at countably many R, it is most sensible to normalize ψ ()=.

26 94 J. Marklof ψ (R) dr = µ(m θ ) µ{(z,φ ;z,φ ) F : 4π f(z,φ ;z,φ ) >R}dR = 4π 5 f d µ = f(t,t ) d t= ψ(r) dr, (76) M π R θ by virtue of relation (35). Proof of Theorem 6.3. It follows from the proof of Theorem 6. that for all but countably many R (recall the definition of the fundamental region F in (3)) ψ (R) = π 4µ{(z,φ ;z,φ ) F : 4π f(z,φ ;z,φ ) >R} = π 4µ{(z,φ ;z,φ ) F : y >y and 4π f (z,φ ;z,φ ) >R}. (77) The asymptotics of ψ (R) is clearly determined by the large values of the theta function f (z,φ ;z,φ ) in the cusps. By Proposition.5 and the refined asymptotic relation f (z,φ ;z,φ )=y /4 f (φ ;z,φ )+O(y ) (78) with the theta function we thus have f (φ ; z,φ )=y /4 n Z f φ,φ (,ny / )e(n x ), ψ (R) = π 4µ{(z,φ ;z,φ ) F : y >y, y > and 4π y/ f (φ ; z,φ ) >R}+O(R ) = π π π 4 min {, F θ y, (4πR) f (φ ; z,φ ) 4 dx dy dφ }dφ y + O(R ). (79) Let us first discuss the integral over the range < (4πR) f (φ ; z,φ ) 4, i.e. (4πR) <y f (φ ;z,φ ) 4, y which, for large R, requires y to be large as well. From the asymptotic relation { y /4 f (φ ; z,φ )= f φ,φ (, ) + O(y ) (y > ) O(v ) (v > ), (8) compare Proposition.5, it follows that the integral over the range in concern is bounded from above and below by the same integral over the ranges y f φ,φ (, ) > 4πR C M R M (for any M> and a suitable constant C M ) which now can be worked out to give, for R large enough,

27 Spectral form factors of rectangle billiards 95 π 4 π π (4πR C M R M )/ f φ,φ (,) π = π 4 π (4πR C M R M )/ f φ,φ (,) π = π 4 (4πR) π min { y, }dx dy dφ y dx dy dφ y 3 dφ dφ f φ,φ (, ) 4 dφ dφ + O M (R M ). (8) As to the remaining range (4πR) >y f (φ ;z,φ ) 4, the same reasoning as before permits to give the upper and lower bounds π π π 4 F θ ((4πR±C M R M )/ f φ,φ (,) ) min {, (4πR) f (φ ; z,φ ) 4 } dx dy dφ y dφ (8) with the truncated fundamental region F θ (T )={z F θ : y<t}. The above integral, restricted to the range < (4πR) f (φ ; z,φ ) 4, equals π π dy π 4 ( ) y dφ dφ + O(R )=O(R ), (83) where the range ( ) of integration of the inner integral is (4πR) f φ,φ (, ) 4 y 4πR f φ,φ (, ). In order to work out the integral over the range > (4πR) f (φ ; z,φ ) 4 let us define the truncated function { H f (φ ; z,φ )= f (φ ;z,φ ) 4 y f φ,φ (, ) 4 for y > f (φ ; z,φ ) 4 otherwise, which is rapidly decreasing in all cusps. The integral over the range under consideration can now be expressed as π π 8π 6 R H f (φ ; z,φ ) dx dy dφ dφ F θ + 8π 6 R y π π 4πR/ fφ,φ (,) The second integral yields π π 8π 6 R log R f φ,φ (, ) 4 dφ dφ + 8π 6 R π π y f φ,φ (, ) 4 dy y dφ dφ (84) + O(R ). (85) f φ,φ (, ) 4 log ( 4π/ f φ,φ (, ) ) dφ dφ. (86)

28 96 J. Marklof Collecting all leading-order terms in the above estimates, we obtain ψ (R) = { π π } 8π 6 f φ,φ (, ) 4 dφ dφ R log R + { π π 8π 6 f φ,φ (, ) 4 log ( 4π/ f φ,φ (, ) ) dφ dφ π π + H f (φ ; z,φ ) dx dy dφ dφ F θ π π + } f φ,φ (, ) 4 dφ dφ R + O(R ). (87) The coefficients c ψ and d ψ are thus determined. By virtue of (3) we have, after some change of variables, π π f φ,φ (, ) 4 dφ dφ = 4 e(t R u +t u )f(t,t )d t 4 d u,(88) R which concludes the proof of Theorem 6.3. y 6.3. The it distribution general cut-off functions. We shall now assume only that f (and thus ψ) is piecewise continuous and of compact support. Theorem 6.5. Let ψ be piecewise continuous and of compact support. Then there exists a decreasing function : R + R + with ψ ()=, ψ (R)dR = discontinuous only for at most countably many R, such that, ψ(r) dr, Probρ,σ {K,ψ (,)>R}= ψ (R), except possibly at the discontinuities of ψ (R). Proof. The proof follows along the lines of the proof of Theorem 4 in [46]. For a given ɛ> choose a function ψ ɛ C (R + ) of compact support, such that ψ(r) ψ ɛ (r) dr < ɛ. (89) The crucial observation is that, for large enough, we have K (α),ψ ψ ɛ (τ,)dτ dα Cɛ, (9) I A for some constant C independent of and ɛ. This fact is a consequence of Proposition 6. and Condition (89). Consider the set S ɛ y = {τ I,α A : K (α),ψ ψ ɛ (τ,) <ɛ / }.

29 Spectral form factors of rectangle billiards 97 The integral over the complement of this set must satisfy Cɛ > K (α),ψ ψ ɛ (τ,)dτ dα hence (I A) S ɛ y (I A) S ɛ y ɛ / dτ dα, S ɛ y > I A Cɛ /. (9) Define the distribution function { ifk (α),ψ (τ,) >R D R,ψ (τ,α,)= otherwise, (9) and the probability ψ,y (R) = By virtue of (9) we have the inclusions I A D R,ψ (τ,α,)ρ(τ) σ(α)dτ dα. (93) ψɛ,y(r + ɛ / ) C ɛ / ψ,y (R) ψɛ,y(r ɛ / )+C ɛ /, (94) where C does not depend on ɛ, y, R. By Theorem 6., for y, the left and right hand side have the its ψ y ɛ,y(r ± ɛ / )= ψɛ (R±ɛ / ) (95) except for countably many R, ɛ. Some analysis shows (for details compare [46]) that for every δ>there is an ɛ>such that ψɛ (R ɛ / ) ψɛ (R + ɛ / ) <δ (96) (except for countably many R). Hence there is a function ψ (R) such that y ψ,y(r) = ψ (R), (97) which proves the claim Random walks. Proof of Theorem.. The task is to show that the difference between { Prob ρ,σ N } e( (α) j τ) N >R and Prob ρ,σ { N (α) j j= (α) e(j τ) N } >R vanishes for N, since then Theorem. will follow from Theorem 6.5. First notice that

30 98 J. Marklof N N j= e( (α) j τ) N (α) e(j τ) N (α) j = N j X (α) N j τ), (98) e( (α) where X (α) N is the set of j defined by X (α) N = {j N : (α) j >N} {j>n : (α) j N}. From the asymptotic relation () it can be readily seen that the number of elements in X (α) N is bounded by #X (α) N N, (99) where the implied constant does not depend on α A (A fixed). It is therefore clear from the calculation done to prove Proposition 6. that e( (α) j τ) dτ dα () I A N j X (α) N vanishes for large N. Hence we can use the inclusion principle in the same fashion as in the proof of Theorem 6.5 (the vanishing of () is the analogue of relation (9)) to prove the existence of the it (R), which furthermore has to equal (R) = χ (R), with χ the characteristic function of the interval [, ]. 7. Rational α The simplest case is α =, since z = z and so the form factor is related to the theta function f (z,φ;z,φ)by ˆK (),ψ (τ,)= 4π f(z,; z,). The function F (z,φ) = f (z,φ;z,φ) can now be viewed as a function on the manifold M θ, which is embedded as a three-dimensional submanifold in M θ, and we can apply the theorems which were developed in the beginning of Sect. 5. This observation holds in a similar way for all rational α = p q ; however, the corresponding embedded threemanifold M p becomes densely distributed in M q θ when the sequence of rationals p q approaches an irrational. Let us discuss this in more detail. For a given integer N we define the congruence subgroups Ɣ (N)ofSL(,Z)by {( ) } ab Ɣ (N)= SL(, Z) : c mod N. () cd With we find [58] that η N = ( ) N Ɣ (N) =η N SL(, Z)η N SL(, Z). ()