on the connection between irregular trajectories and the distribution of quantum level spacings for a model system the distribution of spacings between adjacent levels in specified intervals of the energy spectrum is compared with a semiclassical distribution this distribution berry and robnik 1984 depends on the size of the phase space volume filled with irregular trajectories good agreement is found between quantal and semiclassical spacing distributions in particular the transition from regularity to irregularity observed in the quantum calculation haller et al 1984 is well reproduced by the semiclassical results

coupled modes semiclassical treatment of nonadiabatic transitions we analyse the schrodinger wave equation of a two level or spinorial hamiltonian from a classical point of view an iterative scheme the coupled mode semiclassical formalism is proposed allowing us to deal with the nonadiabatic transfer as the wkb expansion it allows the one dimensional schrodinger equation to be integrated by successive quadratures finally we show that time dependent information can be drawn from the previous purely stationary analysis by extending the notion of group velocity the proposed formalism is thus coherent with an image of multiple trajectories conforming more to physical behaviour than a single trajectory

diagonal approximation of the form factor of the unitary group the form factor of the unitary group u n endowed with the haar measure characterizes the correlations within the spectrum of a typical unitary matrix it can be decomposed into a sum over pairs of periodic orbits where by periodic orbit we understand any sequence of matrix indices from here the diagonal approximation can be defined in the usual fashion as a sum only over pairs of identical orbits we prove that as we take the dimension n to infinity the diagonal approximation becomes exact that is converges to the full form factor in the interval 0 1

distribution of eigenvalues for the modular group the two point correlation functions of energy levels for free motion on the modular domain both with periodic and dirichlet boundary conditions are explicitly computed using a generalization of the hardy littlewood method it is shown that in the limit of small separations they show an uncorrelated behaviour and agree with the poisson distribution but they have prominent number theoretical oscillations at larger scale the results agree well with numerical simulations

resummation and the semiclassical theory of spectral statistics we address the question as to why in the semiclassical limit classically chaotic systems generically exhibit universal quantum spectral statistics coincident with those of random matrix theory to do so we use a semiclassical resummation formalism that explicitly preserves the unitarity of the quantum time evolution by incorporating duality relations between short and long classical orbits this allows us to obtain both the non oscillatory and the oscillatory contributions to spectral correlation functions within a unified framework thus overcoming a significant problem in previous approaches in addition our results extend beyond the universal regime to describe the system specific approach to the semiclassical limit

maximum norms of chaotic quantum eigenstates and random waves abstract the growth of the maximum norms of quantum eigenstates of classically chaotic systems with increasing energy is investigated the maximum norms provide a measure for localization effects in eigenfunctions an upper bound for the maxima of random superpositions of random functions is derived for the random wave model this gives the bound c ln e in the semiclassical limit e the growth of the maximum norms of random waves is compared with the growth of the maximum norms of the eigenstates of six quantum billiards which are classically chaotic the maximum norms of these systems are numerically shown to be conform with the random wave model furthermore the distribution of the locations of the maximum norms is discussed

energy level statistics in the transition region between integrability and chaos a generic one parameter family of billiards discovered and introduced by robnik 1983 is used to study the spectral properties of corresponding quantum systems when the parameter is varied a smooth transition from an integrable system over a typical kam system to an almost ergodic system can be observed the authors calculate up to 7600 lowest reliable energy levels a detailed analysis of the numerical data shows significant deviation from the semiclassical berry robnik formulae for the nearest neighbour level spacing distribution p s except for large level spacings s 1 which can only be explained by a very slow convergence towards the semiclassical regime where these formulae are predicted to be correct at small s the power law level repulsion is clearly observed and a fit by the phenomenological formula by izrailev 1988 1989 is statistically significant

transition from poissonian to gaussian orthogonal ensemble level statistics in a modified artin s billiard one wall of an artin s billiard on the poincar e half plane is replaced by a one parameter mathit c mathit p family of nongeodetic walls a brief description of the classical phase space of this system is given in the quantum domain the continuous and gradual transition from the poisson like to gaussian orthogonal ensemble goe level statistics due to the small perturbations breaking the symmetry responsible for the arithmetic chaos at mathit c mathit p 1 is studied another goe ensuremath rightarrow poisson transition due to the mixed phase space for large perturbations is also investigated a satisfactory description of the intermediate level statistics by the brody distribution was found in both cases the study supports the existence of a scaling region around mathit c mathit p 1 a finite size scaling relation for the brody parameter as a function of 1 mathit c mathit p and the number of levels considered can be established

partner orbits and action differences on compact factors of the hyperbolic plane ii higher order encounters abstract physicists have argued that periodic orbit bunching leads to universal spectral fluctuations for chaotic quantum systems to establish a more detailed mathematical understanding of this fact it is first necessary to look more closely at the classical side of the problem and determine orbit pairs consisting of orbits which have similar actions we specialize to the geodesic flow on compact factors of the hyperbolic plane as a classical chaotic system the companion paper huynh and kunze 2015 proved the existence of a unique periodic partner orbit for a given periodic orbit with a small angle self crossing in configuration space that is a 2 encounter and derived an estimate for the action difference of the orbit pair in this paper we provide an inductive argument to deal with higher order encounters we prove that a given periodic orbit including an l parallel encounter has l 1 1 partner orbits we construct partner orbits and give estimates for the action differences between orbit pairs

quantising a generic family of billiards with analytic boundaries a generic family of plane billiards has been discovered recently the shape of the boundary is given by the quadratic conformal image of the unit circle and is thus real analytic for small deformations of the unit disc the billiard is a typical kam system but becomes ergodic or even mixing when the curvature of the boundary vanishes at some point the kolmogorov entropy has been calculated and it increases with the deformation of the boundary the author studies aspects of the quantum chaos for this billiard he solves numerically the eigenvalue problem for the laplace operator with dirichlet s boundary condition he examines the spectrum and inspects the avoided crossings at which mixing of nearby states occurs the variation of the nodal structure and of the localisation properties of the eigenfunctions is studied in analysing the level spacing distribution he finds a continuous transition from the poisson distribution towards the wigner distribution the exponent in the level repulsion law varies continuously along with a generic perturbation for small perturbations it seems to be proportional to the square root of the perturbation parameter

from classical periodic orbits to the quantization of chaos we present a quantitative analysis of selberg s trace formula viewed as an exact version of gutzwiller s semiclassical periodic orbit theory for the quantization of classically chaotic systems two main applications of the trace formula are discussed in detail i the periodic orbit sum rules giving a smoothing of the quantal energy level density ii the selberg zeta function as a prototype of a dynamical zeta function defined as an euler product over the classical periodic orbits and its analytic continuation across the entropy barrier by means of a dirichlet series it is shown how the long periodic orbits can be effectively taken into account by a universal remainder term which is explicitly given as an integral over an orbit selection function numerical results are presented for the free motion of a point particle on compact riemann surfaces hadamard gutzwiller model which is the primary testing ground for our ideas relating quantum mechanics and classical mechanics in the case of strong chaos our results demonstrate clearly the crucial role played by the long periodic orbits an exact rule for quantizing chaos is derived for such systems where the dirichlet series representing the selberg zeta function converges on the critical line explicit formulae are given for the computation of the abscissae of absolute and conditional convergence respectively of these dynamical dirichlet series for the two riemann surfaces considered it turns out that one can cross the entropy barrier but that the critical line cannot be reached by a convergent dirichlet series it would seem that this is the main reason why the riemann siegel lookalike formula recently conjectured by m v berry and j p keating fails in generating the lower lying quantal energies for these strongly chaotic systems

on fundamental domains of arithmetic fuchsian groups let k be a totally real algebraic number field and o an order in a quaternion algebra a over k assume that the group o 1 of units in o with reduced norm equal to 1 is embedded into psl 2 r as an arithmetic fuchsian group it is shown how ford s algorithm can be effectively applied in order to determine a fundamental domain of o 1 as well as a complete system of generators of o 1

leading off diagonal approximation for the spectral form factor for uniformly hyperbolic systems we consider the semiclassical approximation to the spectral form factor k for two dimensional uniformly hyperbolic systems with time reversal symmetry and derive the first off diagonal correction for small the result agrees with the 2 term of the form factor for the goe random matrix ensemble

diabolical points in the spectra of triangles accidental degeneracies between energy levels e j and e j 1 of a real hamiltonian can occur generically in a family of hamiltonians labelled by at least two parameters x y energy level surfaces in e x y space have locally a double cone diabolo connection and we refer to the degeneracies themselves as diabolical points we studied the family of systems in which a particle moves freely within hard walled triangles vibrations of triangular membranes with x and y labelling two of the angles using an efficient green function technique to compute the levels we found several diabolical points for low lying levels as well as some symmetry degeneracies the lowest diabolical point occurred for levels 5 and 6 of the triangle 130 57 30 73 18 70 the conical structure was confirmed by noting that the normal derivative u of the wavefunction at a boundary point changed sign during a small circuit of the diabolical point the form of the variation of u around a circuit and the changing pattern of nodal lines of agreed with theoretical expectations an estimate of the total number of degeneracies n d j involving levels 1 through j based on the energy scaling of cone angles and the level spacing distribution gave n d j j 2 5 and our limited data support this prediction analytical theory confirmed that for thin triangles where our computational method is slow there are no degeneracies in the energy range studied

semiclassical theory of spectral rigidity the spectral rigidity l of a set of quantal energy levels is the mean square deviation of the spectral staircase from the straight line that best fits it over a range of l mean level spacings in the semiclassical limit 0 formulae are obtained giving l as a sum over classical periodic orbits when l l max where l max n 1 for a system of n freedoms l is shown to display the following universal behaviour as a result of properties of very long classical orbits if the system is classically integrable all periodic orbits filling tori l 1 5 l as in an uncorrelated poisson eigenvalue sequence if the system is classically chaotic all periodic orbits isolated and unstable and has no symmetry l in l 2 2 d if 1 l l max as in the gaussian unitary ensemble of random matrix theory if the system is chaotic and has time reversal symmetry l in l 2 e if 1 l l max as in the gaussian orthogonal ensemble when l l max l saturates non universally at a value determined by short classical orbits of order n 1 for integrable systems and in 1 for chaotic systems these results are obtained by using the periodic orbit expansion for the spectral density together with classical sum rules for the intensities of long orbits and a semiclassical sum rule restricting the manner in which their contributions interfere for two examples l is studied in detail the rectangular billiard integrable and the riemann zeta function assuming its zeros to be the eigenvalues of an unknown quantum system whose unknown classical limit is chaotic

energy level statistics of the hadamard gutzwiller ensemble abstract the statistical properties of the quantal energy levels of the hadamard gutzwiller ensemble whose classical members belong to the class of systems with hard chaos are investigated based on a sample of 4500 energy levels it is shown that the short range statistics as nearest neighbour spacing distributions are governed by the goe predictions of random matrix theory which was first surmised by wigner and by landau and smorodinsky for nuclear level statistics this result strengthens the hypothesis that quantum systems with chaotic classical counterpart display level repulsion as predicted by random matrix theory however the level statistics describing correlations over greater level distances deviate from the goe predictions which is explained as a simple consequrnce of the fact that the spectral rigidity 3 l introduced by dyson and mehta saturates non universally at a finite value for l in complete agreement with the semiclassical theory developed by berry

exact theory for the quantum eigenstates of a strongly chaotic system we present an exact theory for the quantum eigenstates of a strongly chaotic system which consists of a point particle sliding freely on a two dimensional compact surface of constant negative curvature the main result is a general sum rule which relates an almost arbitrarily smoothed sum over the quantum wavefunctions to a sum over the lengths of the classical orbits as an example we apply this formula to the gaussian smoothed sum over wavefunctions the question of scars as imprints of individual closed orbits is discussed and the connection with bogomolny s semiclassical theory is explicitly worked out the investigation of the amplitude distribution p reveals that the higher excited wavefunctions behave as gaussian random waves

spectral statistics for unitary transfer matrices of binary graphs quantum graphs have recently been introduced as model systems to study the spectral statistics of linear wave problems with chaotic classical limits it is proposed here to generalize this approach by considering arbitrary directed graphs with unitary transfer matrices an exponentially increasing contribution to the form factor is identified when performing a diagonal summation over periodic orbit degeneracy classes a special class of graphs so called binary graphs is studied in more detail for these the conditions for periodic orbit pairs to be correlated including correlations due to the unitarity of the transfer matrix can be given explicitly using combinatorial techniques it is possible to perform the summation over correlated periodic orbit pair contributions to the form factor for some low dimensional cases gradual convergence towards random matrix results is observed when increasing the number of vertices of the binary graphs

an algorithm for the computation of eigenvalues spectral zeta functions and zeta determinants on hyperbolic surfaces we present a rigorous scheme that makes it possible to compute eigenvalues of the laplace operator on hyperbolic surfaces within a given precision the method is based on an adaptation of the method of particular solutions to the case of locally symmetric spaces and on explicit estimates for the approximation of eigenfunctions on hyperbolic surfaces by certain basis functions it can be applied to check whether or not there is an eigenvalue in an e neighborhood of a given number 0 this makes it possible to find all the eigenvalues in a specified interval up to a given precision with rigorous error estimates the method converges exponentially fast with the number of basis functions used combining the knowledge of the eigenvalues with the selberg trace formula we are able to compute values and derivatives of the spectral zeta function again with error bounds as an example we calculate the spectral determinant and the casimir energy of the bolza surface and other surfaces

a multishift hessenberg method for pole assignment of single input systems a new algorithm is proposed for the pole assignment of single input linear time invariant systems the proposed algorithm belongs to the family of hessenberg methods and is based on an implicit multishift qr like technique the new method compares favourably in many respects speed memory usage with existing numerically stable methods its improved vectorizability guarantees good opportunities for parallel implementation on high performance computers

semiclassical level spacings when regular and chaotic orbits coexist the authors calculate semiclassical limiting level spacing distributions p s for systems whose classical energy surface is divided into a number of separate region in which motion is regular or chaotic in the calculation it is assumed that the spectrum is the superposition of statistically independent sequences of levels from each of the classical phase space regions sequences from regular regions having poisson distributions and those from irregular regions having wigner distributions the formulae for p s depend on the sum of the liouville measures of all the classical regular regions and on the separate liouville measures of the significant chaotic regions

periodic orbits and semiclassical quantization of dispersing billiards periodic orbits in a dispersing billiard system consisting of three circular arcs are studied numerically by using a partial coding rule together with an efficient method for enumerating periodic orbits on the real billiard plane by examining several statistical measures it is shown that the length spectrum and the stability exponents are highly uncorrelated the validity of the semiclassical trace formula is also tested and a remarkable agreement of the semiclassical and quantum density of states is obtained at least for about the lower 15 levels

the selberg trace formula for dirac operators we examine spectra of dirac operators on compact hyperbolic surfaces particular attention is devoted to symmetry considerations leading to nontrivial multiplicities of eigenvalues the relation to spectra of maas laplace operators is also exploited our main result is a selberg trace formula for dirac operators on hyperbolic surfaces

an algorithm to assign eigenvalues in a hessenberg matrix single input case we propose a conceptually simple algorithm to assign eigenvalues in a hessenberg matrix the method is based on the evaluation of a simple recursive relation and it is extremely easy to program on a computer numerical experiments compare favorably to those of other methods recently proposed for the problem

on the rate of quantum ergodicity on hyperbolic surfaces and for billiards abstract the rate of quantum ergodicity is studied for three strongly chaotic anosov systems the quantal eigenfunctions on a compact riemannian surface of genus g 2 and of two triangular billiards on a surface of constant negative curvature are investigated one of the triangular billiards belongs to the class of arithmetic systems there are no peculiarities observed in the arithmetic systems concerning the rate of quantum ergodicity this contrasts to the peculiar behaviour with respect to the statistical properties of the quantal levels it is demonstrated that the rate of quantum ergodicity in the three considered systems fits well with the known upper and lower bounds furthermore sarnak s conjecture about quantum unique ergodicity for hyperbolic surfaces is confirmed numerically in these three systems

periodic orbit sum rules for the hadamard gutzwiller model abstract it is shown how a variety of periodic orbit sum rules can be used to extract information about a quantum mechanical system whose classical counterpart is completely chaotic from knowledge only of the classical system and vice versa the basis is the selberg trace formula an exact analogue for the hadamard gutzwiller model of the semiclassical periodic orbit theory of gutzwiller which relates the quantal energies to the lengths of the periodic orbits of the classical system statistical properties of the quantal energies in the low energy region are studied where we restrict ourselves to the level spacing and spectral rigidity

semiclassical limits for the hyperbolic plane we study the concentration properties of high energy eigenfunctions for the laplace beltrami operator of the hyperbolic plane with special consideration of automorphic eigenfunctions at the center of our investigation is the microlocal lift of an eigenfunction to sl 2 introduced by s zelditch the microlocal lift is based on s helgason s fourier transform and has a straightforward description in terms of the lie algebra sl 2 we begin with an elementary demonstration of zelditch s exact differential equation for the microlocal lift further for a sequence of suitably bounded eigenfunctions with eigenvalues tending to infinity we show that the limit of microlocal lifts is a geodesic flow invariant positive measure our main consideration is the microlocal lifts of the elementary eigenfunctions constructed from the macdonald bessel functions we find that with a scaling of auxiliary parameters the corresponding high energy limit converges to the positive dirac measure for the lift to sl 2 of a single geodesic on the upper half plane in particular at high energy the microlocal lift of a macdonald bessel function is concentrated along the lift of a single geodesic we further find that the convergence is uniform in parameters and thus our considerations can be applied to study automorphic eigenfunctions in particular the maass cusp forms for certain cofinite subgroups of sl 2 a formula is presented for the microlocal lift of a cusp form as an integral of a family of dirac measures and a measure given directly by sums of products of the classical fourier coefficients of the cusp form the formula establishes that questions on quantum chaos for automorphic eigenfunctions are equivalent to classically stated questions on twisted sums of coefficients in particular we find that the uniform distribution for the microlocal limit is equivalent to a uniform distribution for the limit of twisted coefficient sums we extend the considerations to the case of the nonholomorphic eisenstein series for the modular group we find that a result of w luo and p sarnak is equivalent to a limit sum formula involving the riemann zeta function and the elementary divisor function the formula is suggestive of a formula of s ramanujan

the effect of pitchfork bifurcations on the spectral statistics of hamiltonian systems we present a quantitative semiclassical treatment of the effects of bifurcations on the spectral rigidity and the spectral form factor of a hamiltonian quantum system defined by two coupled quartic oscillators which on the classical level exhibits mixed phase space dynamics we show that the signature of a pitchfork bifurcation is two fold beside the known effect of an enhanced periodic orbit contribution due to its peculiar hbar dependence at the bifurcation we demonstrate that the orbit pair born at the bifurcation gives rise to distinct deviations from universality slightly above the bifurcation this requires a semiclassical treatment beyond the so called diagonal approximation our semiclassical predictions for both the coarse grained density of states and the spectral rigidity are in excellent agreement with corresponding quantum mechanical results

an exponentially increasing semiclassical spectral form factor for a class of chaotic systems the spectral form factor k tau plays a crucial role in the understanding of the statistical properties of quantal energy spectra of strongly chaotic systems in terms of periodic orbits it allows the semiclassical computation of those statistics that are bilinear in the spectral density d e like the spectral rigidity delta 3 l and the number variance sigma 2 l since berry s 1985 1991 work on the semiclassical approximation of the spectral rigidity in terms of periodic orbits it is generally assumed that the periodic orbit expression for the spectral form factor universally obeys k tau 1 for tau 1 we show that for a wide class of strongly chaotic systems including billiards with neumann boundary conditions and the motion on some riemann surfaces the asymptotic behaviour of the semiclassical spectral form factor k tau depends very sensitively on the averaging employed a gaussian averaging is preferable from a theoretical as well as from a numerical point of view to for example a rectangular averaging however we show in this paper that the gaussian averaging leads in some cases to an asymptotic behaviour like k tau approximately ec tau where c 0 depends only on the energy e at which the statistic is considered

arithmetical chaos and violation of universality in energy level statistics a class of strongly chaotic systems revealing a strange arithmetical structure is discussed whose quantal energy levels exhibit level attraction rather than repulsion as an example the nearest neighbor level spacings for artin s billiard have been computed in a large energy range it is shown that the observed violation of universality has its root in the existence of an infinite number of hermitian operators hecke operators which commute with the hamiltonian and generate nongeneric correlations in the eigenfunctions

a maximum likelihood approach to joint image registration and fusion both image registration and fusion can be formulated as estimation problems instead of estimating the registration parameters and the true scene separately as in the conventional way we propose a maximum likelihood approach for joint image registration and fusion in this paper more precisely the fusion performance is used as the criteria to evaluate the registration accuracy hence the registration parameters can be automatically tuned so that both fusion and registration can be optimized simultaneously the expectation maximization algorithm is employed to solve this joint optimization problem the cramer rao bound crb is then derived our experiments use several types of sensory images for performance evaluation such as visual images ir thermal images and hyperspectral images it is shown that the mean square error of estimating the registration parameters using the proposed method is close to the crbs at the mean time an improved fusion performance can be achieved in terms of the edge preservation measure qab f compared to the laplacian pyramid fusion approach

moments of the wigner delay times the wigner time delay is a measure of the time spent by a particle inside the scattering region of an open system for chaotic systems the statistics of the individual delay times whose average is the wigner time delay are thought to be well described by random matrix theory here we present a semiclassical derivation showing the validity of random matrix results in order to simplify the semiclassical treatment we express the moments of the delay times in terms of correlation functions of scattering matrices at different energies in the semiclassical approximation the elements of the scattering matrix are given in terms of the classical scattering trajectories requiring one to study correlations between sets of such trajectories we describe the structure of correlated sets of trajectories and formulate the rules for their evaluation to the leading order in inverse channel number this allows us to derive a polynomial equation satisfied by the generating function of the moments along with showing the agreement of our semiclassical results with the moments predicted by random matrix theory we infer that the scattering matrix is unitary to all orders in the semiclassical approximation

spectral correlations of individual quantum graphs we investigate the spectral properties of chaotic quantum graphs we demonstrate that the energy average over the spectrum of individual graphs can be traded for the functional average over a supersymmetric nonlinear model action this proves that spectral correlations of individual quantum graphs behave according to the predictions of wigner dyson random matrix theory we explore the stability of the universal random matrix behavior with regard to perturbations and discuss the crossover between different types of symmetries

an algorithm for pole assignment of time invariant linear systems an algorithm is suggested for the computation of a linear state feedback for a single input system such that the resultant closed loop system has specified eigenvalues the algorithm is more efficient than many comparable techniques and has some desirable numerical properties it is closely related to the qr algorithm for the eigenproblem and extends to descriptor systems

multimodality image registration by maximization of mutual information a new approach to the problem of multimodality medical image registration is proposed using a basic concept from information theory mutual information mi or relative entropy as a new matching criterion the method presented in this paper applies mi to measure the statistical dependence or information redundancy between the image intensities of corresponding voxels in both images which is assumed to be maximal if the images are geometrically aligned maximization of mi is a very general and powerful criterion because no assumptions are made regarding the nature of this dependence and no limiting constraints are imposed on the image content of the modalities involved the accuracy of the mi criterion is validated for rigid body registration of computed tomography ct magnetic resonance mr and photon emission tomography pet images by comparison with the stereotactic registration solution while robustness is evaluated with respect to implementation issues such as interpolation and optimization and image content including partial overlap and image degradation our results demonstrate that subvoxel accuracy with respect to the stereotactic reference solution can be achieved completely automatically and without any prior segmentation feature extraction or other preprocessing steps which makes this method very well suited for clinical applications

conjugacy classes of g 2 and spectral rigidity the free group 2 is generated by a 1 2 0 1 and b 10 21 and setting x a exp 2 i x b exp 2 i defines a unitary character on 2 for 0 1 a program is devised to compute tr conj class summed over all primitive conjugacy classes of 2 of trace tr combined with a luo sarnak theorem this yields lower bounds for the spectral variance for a large sampling of characters in the range 0 1 the results indicate that the berry conjecture for spectral rigidity does not hold for this set of classically chaotic systems the program is also used to compute x ln n summed over all primitive conjugacy classes of 2 of norm n x the function 0 x is asymptotic to x and the remainder can be written as x x x the values of x are computed for all traces between 3202 and 4802 here x tr 2 2 the s cluster around 0 6 attaining a maximum of 2 3 finally it is proved that the remainder 0 x x has a negative bias by showing that the mean normalized remainder converges to a negative limit

a computational algorithm for pole assignment of linear single input systems an efficient computational algorithm for pole assignment of linear multiinput systems is proposed a preliminary stage of the algorithm is a reduction of the system matrices into orthogonal canonical form the gain matrix elements are then found by orthogonal transformation of the closed loop system matrix into upper quasi triangular form whose diagonal blocks correspond to the desired poles the algorithm is numerically stable the computed gain matrix being exact for a system with slightly perturbed matrices it works equally well with real and complex distinct and multiple poles and is applicable to ill conditioned and high order problems

level statistics in arithmetical and pseudo arithmetical chaos we investigate a long standing riddle in quantum chaos posed by certain fully chaotic billiards with constant negative curvature whose periodic orbits are highly degenerate in length depending on the boundary conditions for the quantum wavefunctions the energy spectra either have uncorrelated levels usually associated with classical integrability or conform to the universal wigner dyson type although the classical dynamics in both cases is the same the resolution turns out surprisingly simple the maslov indices of orbits within multiplets of degenerate length either yield equal phases for the respective feynman amplitudes and thus poissonian level statistics or give rise to amplitudes with uncorrelated phases leading to wigner dyson level correlations the recent semiclassical explanation of spectral universality in quantum chaos is thus extended to the latter case of pseudo arithmetical chaos

universal spectral form factor for chaotic dynamics we consider the semiclassical limit of the spectral form factor k of fully chaotic dynamics starting from the gutzwiller type double sum over classical periodic orbits we set out to recover the universal behaviour predicted by random matrix theory both for dynamics with and without time reversal invariance for times smaller than half the heisenberg time th f 1 we extend the previously known expansion to include the cubic term beyond confirming the random matrix behaviour of individual spectra the virtue of that extension is that the diagrammatic rules come in sight which determine the families of orbit pairs responsible for all orders of the expansion

flux quantization and quantum mechanics on riemann surfaces in an external magnetic field the authors investigate the possibility of applying an external constant magnetic field to a quantum mechanical system consisting of a particle moving on a compact or non compact two dimensional manifold of constant negative gaussian curvature and of finite volume for the motion on compact riemann surfaces they find that a consistent formulation is only possible if the magnetic flux is quantized as it is proportional to the integrated first chern class of a certain complex line bundle over the manifold in the case of non compact surfaces of finite volume they obtain the striking result that the magnetic flux has to vanish identically due to the theorem that any holomorphic line bundle over a non compact riemann surface is holomorphically trivial

full counting statistics of chaotic cavities from classical action correlations we present a trajectory based semiclassical calculation of the full counting statistics of quantum transport through chaotic cavities in the regime of many open channels our method to obtain the mth moment of the density of transmission eigenvalues requires two correlated sets of m classical trajectories therefore generalizing previous works on conductance and shot noise the semiclassical results agree for all values of m with the corresponding predictions from random matrix theory

on the form factor for the unitary group we study the combinatorics of the contributions to the form factor of the group u n in the large n limit this relates to questions about semiclassical contributions to the form factor of quantum systems described by the unitary ensemble

a detailed investigation of the spectral rigidity for integrable hamiltonian systems the authors carry out a detailed analysis of the spectral rigidity for integrable systems with a view towards understanding the deviations from the universal behaviour l 15 in systems with degeneracies in orbit actions in the process they also investigate the saturation region their studies show that the energy dependence of delta 3 infinity is a better indicator of the underlying universality in integrable systems

on invariant 2 2 β ensembles of random matrices abstract we introduce and solve exactly a family of invariant 2 2 random matrices depending on one parameter and we show that rotational invariance and real dyson index are not incompatible properties the probability density for the entries contains a weight function and a multiple trace trace interaction term which corresponds to the representation of the vandermonde squared coupling on the basis of power sums as a result the effective dyson index eff of the ensemble can take any real value in an interval two weight functions gaussian and non gaussian are explored in detail and the connections with ensembles of dumitriu edelman and the so called poisson wigner crossover for the level spacing are respectively highlighted a curious spectral twinning between ensembles of different symmetry classes is unveiled as a consequence the identification between symmetry group orthogonal unitary or symplectic and the exponent of the vandermonde 1 2 4 is shown to be potentially deceptive the proposed technical tool more generically allows for designing actual matrix models which i are rotationally invariant ii have a real dyson index eff iii have a pre assigned confining potential or alternatively level spacing profile the analytical results have been checked through numerical simulations with an excellent agreement eventually we discuss possible generalizations and further directions of research

likelihood maximization approach to image registration a likelihood maximization approach to image registration is developed in this paper it is assumed that the voxel values in two images in registration are probabilistically related the principle of maximum likelihood is then exploited to find the optimal registration the likelihood that given image f one has image g and given image g one has image f is optimized with respect to registration parameters all voxel pairs in the overlapping volume or a portion of it can be used to compute the likelihood a knowledge based method and a self consistent technique are proposed to obtain the probability relation in the knowledge based method prior knowledge of the distribution of voxel pairs in two registered images is assumed while such knowledge is not required in the self consistent method the accuracy and robustness of the likelihood maximization approach is validated by single modality registration of single photon emission computed tomographic spect images and magnetic resonance mr images and by multimodality registration mr spect the results demonstrate that the performance of the likelihood maximization approach is comparable to that of the mutual information maximization technique finally the relationship between the likelihood approach and the entropy conditional entropy and mutual information approaches is discussed

energy level statistics of an integrable billiard system in a rectangle in the hyperbolic plane a separable billiard system in the hyperbolic strip a particular realization of the hyperbolic plane is investigated the author considers a rectangle bounded by straight lines in the hyperbolic strip as an approximation of either a regular octagon with area a 4 pi or of a hyperbolic rectangle bounded by geodesics respectively due to the simple geometry of the approximation the schrodinger equation is separable and leads to simple transcendental equations for the eigenvalues in terms of legendre functions the statistical properties of the eigenvalue spectrum are investigated and checked with respect to weyl s 1912 law level spacing number variance and spectral rigidity respectively in particular the numerical results for the spectral rigidity are in close agreement with the semi classical theory of berry 1985

statistics of resonances for a class of billiards on the poincare half plane the lower boundary of artin s billiard on the poincar half plane is continuously deformed to generate a class of billiards with classical dynamics varying from fully integrable to completely chaotic the quantum scattering problem in these open billiards is described and the statistics of both real and imaginary parts of the resonant momenta are investigated the evolution of the resonance positions is followed as the boundary is varied which leads to large changes in their distribution the transition to arithmetic chaos in artin s billiard which is responsible for the poissonian level spacing statistics of the bound states in the continuum cusp forms at the same time as the formation of a set of resonances all with width and real parts determined by the zeros of riemann s zeta function is closely examined regimes are found which obey the universal predictions of random matrix theory rmt as well as exhibiting non universal long range correlations the brody parameter is used to describe the transitions between different regimes

on the distribution of eigenvalues of maass forms on certain moonshine groups in this paper we study both analytically and numerically questions involving the distribution of eigen values of maass forms on the moonshine groups 0 n where n 1 is a square free integer after we prove that 0 n has one cusp we compute the constant term of the associated non holomorphic eisenstein series we then derive an average weyl s law for the distribution of eigenvalues of maass forms from which we prove the classical weyl s law as a special case the groups corresponding to n 5 and n 6 have the same signature however our analysis shows that asymptotically there are infinitely more cusp forms for 0 5 than for 0 6 we view this result as being consistent with the phillips sarnak philosophy since we have shown unconditionally the existence of two groups which have different weyl s laws in addition we employ hejhal s algorithm together with recently developed refinements from 32 and numerically determine the first 3557 of 0 5 and the first 12474 eigenvalues of 0 6 with this information we empirically verify some conjectured distributional properties of the eigenvalues

semiclassical universality of parametric spectral correlations we consider quantum systems with a chaotic classical limit that depends on an external parameter and study correlations between the spectra at different parameter values in particular we consider the parametric spectral form factor k x which depends on a scaled parameter difference x for parameter variations that do not change the symmetry of the system we show by using semiclassical periodic orbit expansions that the small expansion of the form factor agrees with random matrix theory for systems with and without time reversal symmetry

combinatorial theory of the semiclassical evaluation of transport moments i equivalence with the random matrix approach to study electronic transport through chaotic quantum dots there are two main theoretical approachs one involves substituting the quantum system with a random scattering matrix and performing appropriate ensemble averaging the other treats the transport in the semiclassical approximation and studies correlations among sets of classical trajectories there are established evaluation procedures within the semiclassical evaluation that for several linear and non linear transport moments to which they were applied have always resulted in the agreement with random matrix predictions we prove that this agreement is universal any semiclassical evaluation within the accepted procedures is equivalent to the evaluation within random matrix theory the equivalence is shown by developing a combinatorial interpretation of the trajectory sets as ribbon graphs maps with certain properties and exhibiting systematic cancellations among their contributions remaining trajectory sets can be identified with primitive palindromic factorisations whose number gives the coefficients in the corresponding expansion of the moments of random matrices the equivalence is proved for systems with and without time reversal symmetry

periodic orbits on the regular hyperbolic octagon abstract the length spectrum of closed geodesics on a compact riemann surface corresponding to a regular octagon on the poincare disc is investigated the general form of the elements of the octagon group a discrete subgroup of nu su 1 1 1 in terms of 2 2 matrices is derived and the previously conjectured law for the length of periodic orbits is proved analytically an algorithm for the multiplicity of geodesics with a given length is developed which leads to an efficient enumeration of the periodic orbits of this strongly chaotic system

the spin contribution to the form factor of quantum graphs following the quantization of a graph with the dirac operator spin 1 2 we explain how additional weights in the spectral form factor k due to spin propagation around orbits produce higher order terms in the small asymptotics in agreement with symplectic random matrix ensembles we determine conditions on the group of spin rotations sufficient to generate cse statistics

spectral statistics for quantized skew translations on the torus we study the spectral statistics for quantized skew translations on the torus which are ergodic but not mixing for irrational parameters it is shown explicitly that in this case the level spacing distribution and other common spectral statistics like the number variance do not exist in the semiclassical limit

spectral statistics in chaotic systems with a point interaction we consider quantum systems with a chaotic classical limit that are perturbed by a point like scatterer the spectral form factor k for these systems is evaluated semiclassically in terms of periodic and diffractive orbits it is shown for order 2 and 3 that off diagonal contributions to the form factor which involve diffractive orbits cancel exactly the diagonal contributions from diffractive orbits implying that the perturbation by the scatterer does not change the spectral statistic we further show that parametric spectral statistics for these systems are universal for small changes of the strength of the scatterer

trace formulae for three dimensional hyperbolic lattices and application to a strongly chaotic tetrahedral billiard abstract this paper is devoted to the quantum chaology of three dimensional systems a trace formula is derived for compact polyhedral billiards which tessellate the three dimensional hyperbolic space of constant negative curvature the exact trace formula is compared with gutzwiller s semiclassical periodic orbit theory in three dimensions and applied to a tetrahedral billiard being strongly chaotic geometric properties as well as the conjugacy classes of the defining group are discussed the length spectrum and the quantal level spectrum are numerically computed allowing the evaluation of the trace formula as is demonstrated in the case of the spectral staircase n e which in turn is successfully applied in a quantization condition

form factor for a family of quantum graphs an expansion to third order for certain types of quantum graphs we show that the random matrix form factor can be recovered to at least third order in the scaled time from periodic orbit theory we consider the contributions from pairs of periodic orbits represented by diagrams with up to two self intersections connected by up to four arcs and explain why all other diagrams are expected to give higher order corrections only for a large family of graphs with ergodic classical dynamics the diagrams that exist in the absence of time reversal symmetry sum to zero the mechanism for this cancellation is rather general which suggests that it also applies at higher orders in the expansion this expectation is in full agreement with the fact that in this case the linear contribution the diagonal approximation already reproduces the random matrix form factor for 1 for systems with time reversal symmetry there are more diagrams which contribute at third order we sum these contributions for quantum graphs with uniformly hyperbolic dynamics obtaining 2 3 in agreement with random matrix theory as in the previous calculation of the leading order correction to the diagonal approximation we find that the third order contribution can be attributed to exceptional orbits representing the intersection of diagram classes

a direct algorithm for pole assignment of time invariant multi input linear systems using state feedback abstract a direct algorithm is suggested for the computation of the linear state feedback for multi input systems such that the resultant closed loop system matrix has specified eigenvalues the extra freedom can be used in different ways for example to decrease some norm of the feedback matrix or to improve the condition of some of the eigenvalues of the closed loop matrix the algorithm uses unitary transformations for numerical reliability and is based on ideas from the qr algorithm for solving the eigenproblem the stability of the algorithm is proven by doing a backward rounding error analysis and numerical examples are given as well

multimodality image registration by maximization of quantitative qualitative measure of mutual information this paper presents a novel image similarity measure referred to as quantitative qualitative measure of mutual information q mi for multimodality image registration conventional information measures e g shannon s entropy and mutual information mi reflect quantitative aspects of information because they only consider probabilities of events in fact each event has its own utility to the fulfillment of the underlying goal which can be independent of its probability of occurrence thus it is important to consider both quantitative i e probability and qualitative i e utility measures of information in order to fully capture the characteristics of events accordingly in multimodality image registration q mi should be used to integrate the information obtained from both the image intensity distributions and the utilities of voxels in the images different voxels can have different utilities for example in brain images two voxels can have the same intensity value but their utilities can be different e g a white matter wm voxel near the cortex can have higher utility than a wm voxel inside a large uniform wm region in q mi the utility of each voxel in an image can be determined according to the regional saliency value calculated from the scale space map of this image since the voxels with higher utility values or saliency values contribute more in measuring q mi of the two images the q mi based registration method is much more robust compared to conventional mi based registration methods also the q mi based registration method can provide a smoother registration function with a relatively larger capture range in this paper the proposed q mi has been validated and applied to the rigid registrations of clinical brain images such as mr ct and pet images

maas cusp forms for large eigenvalues we investigate the numerical computation of maa2ss cusp forms for the modular group corresponding to large eigenvalues we present fourier coefficients of two cusp forms whose eigenvalues exceed r 40000 these eigenvalues are the largest that have so far been found in the case of the modular group they are larger than the 130millionth eigenvalue

periodic orbits and a correlation function for the semiclassical density of states through the rule that there is one quantum state per planck cell in phase space classical mechanics supplies an average semiclassical density of states for a system with bound motion can it supply more refined averages such as the correlation function for the density of states i shall argue that for the two extreme cases of integrable classical motion and ergodic classical motion information can indeed be obtained on not quite this but a closely related correlation function the results for the two types of motion are markedly different

crossing the entropy barrier of dynamical zeta functions dynamical zeta functions are an important tool to quantize chaotic dynamical systems the basic quantization rules require the computation of the zeta functions on the real energy axis where their euler product representations running over the classical periodic orbits usually do not converge due to the existence of the so called entropy barrier determined by the topological entropy of the classical system we show that the convergence properties of the dynamical zeta functions rewritten as dirichlet series are governed not only by the well known topological and metric entropy but depend crucially on subtle statistical properties of the maslov indices and of the multiplicities of the periodic orbits that are measured by a new parameter for which we introduce the notion of a third entropy if and only if the third entropy is nonvanishing one can cross the entropy barrier if it exceeds a certain value one can even compute the zeta function in the physical region by means of a convergent dirichlet series a simple statistical model is presented which allows to compute the third entropy four examples of chaotic systems are studied in detail to test the model numerically

multiplicities of periodic orbit lengths for non arithmetic models multiplicities of periodic orbit lengths for non arithmetic hecke triangle groups are discussed it is demonstrated both numerically and analytically that at least for certain groups the mean multiplicity of periodic orbits with exactly the same length increases exponentially with the length the main ingredient used is the construction of a joint distribution of periodic orbits when group matrices are transformed by field isomorphisms the method can be generalized to other groups for which traces of group matrices are integers of an algebraic field of finite degree

semiclassical form factor for spectral and matrix element fluctuations of multidimensional chaotic systems we present a semiclassical calculation of the generalized form factor kab tau which characterizes the fluctuations of matrix elements of the operators a and b in the eigenbasis of the hamiltonian of a chaotic system our approach is based on some recently developed techniques for the spectral form factor of systems with hyperbolic and ergodic underlying classical dynamics and f 2 degrees of freedom that allow us to go beyond the diagonal approximation first we extend these techniques to systems with f 2 then we use these results to calculate kab tau we show that the dependence on the rescaled time tau time in units of the heisenberg time is universal for both the spectral and the generalized form factor furthermore we derive a relation between kab tau and the classical time correlation function of the weyl symbols of a and b

pseudointegrable systems in classical and quantum mechanics abstract we study particles moving in planar polygonal enclosures with rational angles and show by several methods that trajectories in the classical phase space explore two dimensional invariant surfaces which are generically not tori as in integrable systems but instead have the topology of multiply handled spheres the quantum mechanics of one such pseudointegrable system is studied in detail by computing energy levels using an exact formalism this system consists of motion on a unit coordinate torus containing a square reflecting obstacle with side l we find that neighbouring levels avoid degeneracies as l varies and that the probability distribution for the spacing s of adjacent levels vanishes linearly as s 0 level repulsion the weyl area rule plus edge and corner corrections gives a very accurate approximation for the mean level density oscillatory corrections to the mean level density are given as a sum over closed classical paths for pseudointegrable systems these closed paths form families covering part of the phase space invariant surfaces

torsion homology growth and cycle complexity of arithmetic manifolds let m be an arithmetic hyperbolic 3 manifold such as a bianchi manifold we conjecture that there is a basis for the second homology of m where each basis element is represented by a surface of low genus and give evidence for this we explain the relationship between this conjecture and the study of torsion homology growth

selberg super trace formula for super riemann surfaces ii elliptic and parabolic conjugacy classes and selberg super zeta functions further contributions developing a super analogue of the classical selberg trace formula the selberg super trace formula are presented this paper deals with the calculation of contributions arising from elliptic and parabolic conjugacy classes to the selberg super trace formula for super riemann surfaces analytic properties and the functional equation for the corresponding selberg super zeta functionr0 r1 andzs respectively are derived and discussed in particular the elliptic contributions to a super fuchsian group only alter the multiplicities of the trivial zeros and poles of the selberg super zeta functionr0 r1 andzs respectively already due to the hyperbolic conjugacy classes the parabolic conjugacy classes introduce new features in the analytical structure

periodic orbit quantization of bound chaotic systems the authors present periodic orbit quantizations of the hyperbola billiard and the x2y2 potential these two systems may be considered as belonging to the one parameter family of potentials x2y2 1a the quantum states are determined by means of the zeros of an expanded and truncated selberg zeta function the symmetries of the problem are considered and the selberg zeta function is factorized into the irreducible representations of the symmetry group the thus calculated eigenenergies are in good agreement with quantum mechanical calculations and converge when the number of terms in the expansion is increased the results strongly indicate that the trace formula provides individual quantum eigenstates for chaotic systems

lyndon word decompositions and pseudo orbits on q nary graphs abstract a foundational result in the theory of lyndon words words that are strictly earlier in lexicographic order than their cyclic permutations is the chen fox lyndon theorem which states that every word has a unique non increasing decomposition into lyndon words this article extends this factorization theorem obtaining the proportion of these decompositions that are strictly decreasing this result is then used to count primitive pseudo orbits sets of primitive periodic orbits on q nary graphs as an application we obtain a diagonal approximation to the variance of the characteristic polynomial coefficients of q nary quantum graphs

separation of variables in path integrals and path integral solution of two potentials on the poincare upper half plane the author discusses how to separate variables in path integrals it is assumed that a one dimensional problem with potential v x has an exact solution with energy levels elambda and wavefunctions psi lambda in order to perform the separation of variables a time transformation is performed back and forth in the path integral which allows one to insert the path integral solution corresponding to the potential v x the author illustrates the method by discussing some specific potential problems on the poincare upper half plane

action correlations and random matrix theory the correlations in the spectra of quantum systems are intimately related to correlations which are of genuine classical origin and which appear in the spectra of actions of the classical periodic orbits of the corresponding classical systems we review this duality and the semiclassical theory which brings it about the conjecture that the quantum spectral statistics are described in terms of random matrix theory leads to the proposition that the classical two point correlation function is also given in terms of a universal function we study in detail the spectrum of actions of the baker map and use it to illustrate the steps needed to reveal the classical correlations their origin and their relation to symbolic dynamics

hyperbolic octagons and teichmuller space in genus 2 in this paper we study different models of the teichmuller space of compact hyperbolic surfaces with special emphasis on their construction by geodesic octagons such surfaces have become increasingly interesting due to their applications to the physical behavior of chaotic quantum systems they are on the one hand complex enough to show the relevant features and on the other hand they possess a simple mathematical structure allowing practical implementations we give a new description of teichmuller space and the mapping class group in terms of geometric data of the octagons this provides modellings based on the vertices and also on the generators of the associated isometry group in addition we explicitly connect our approach with other models currently used in the literature helling matrices fenchel nielsen parameters the result of the paper may be considered as a computational tool kit to work with a specific model and to relate it with the others

semiclassics for chaotic systems with tunnel barriers the addition of tunnel barriers to open chaotic systems as well as representing more general physical systems leads to a much richer semiclassical dynamics in particular we present here a complete semiclassical treatment for these systems in the regime where ehrenfest time effects are negligible and for times shorter than the heisenberg time to start we explore the trajectory structures which contribute to the survival probability and find results that are also in agreement with random matrix theory then we progress to the treatment of the probability current density and are able to show using recursion relation arguments that the continuity equation connecting the current density to the survival probability is satisfied to all orders in the semiclassical approximation following on we also consider a correlation function of the scattering matrix for which we have to treat a new set of possible trajectory diagrams by simplifying the contributions of these diagrams we show that the results obtained here are consistent with the known properties of the scattering matrix the correlation function can be trivially connected to the ac and dc conductances quantities of particular interest for which finally we present a semiclassical expansion

the generalized eigenstructure problem in linear system theory the algebraic theory of linear time invariant systems has been studied in large detail during the past few decades and numerous computational algorithms have been developed to solve problems arising in this context in this paper the numerical aspects of a certain class of such algorithms dealing with what the author calls generalized eigenstructure problems are discussed some new and or modified algorithms are presented both the nmnerical stability of the algorithms and the conditioning of the problems they solve are analyzed using numerical criteria

the behaviour of resonances in hecke triangular billiards under deformation the right hand boundary of artin s billiard on the poincare half plane is continuously deformed to generate a class of chaotic billiards which includes fundamental domains of the hecke groups 2 n at certain values of the deformation parameter the quantum scattering problem in these open chaotic billiards is described and the distributions of both real and imaginary parts of the resonant eigenvalues are investigated the transitions to arithmetic chaos in the cases n 4 6 are closely examined and the explicit analytic form for the scattering matrix is given together with the fourier coefficients for the scattered wavefunction the n 4 and 6 cases have an additional set of regular equally spaced resonances compared to artin s billiard n 3 for a general deformation a numerical procedure is presented which generates the resonance eigenvalues and the evolution of the eigenvalues is followed as the boundary is varied continuously which leads to dramatic changes in their distribution for deformations away from the non generic arithmetic cases including that of the tiling hecke triangular billiard n 5 the distributions of the positions and widths of the resonances are consistent with the predictions of a random matrix theory

eigenvalue spacings for quantized cat maps according to one of the basic conjectures in quantum chaos the eigenvalues of a quantized chaotic hamiltonian behave like the spectrum of the typical member of the appropriate ensemble of random matrices we study one of the simplest examples of this phenomenon in the context of ergodic actions of groups generated by several linear toral automorphisms cat maps our numerical experiments indicate that for generic choices of cat maps the unfolded consecutive spacing distribution in the irreducible components of the nth quantization given by the n dimensional weil representation approaches the goe gse law of random matrix theory for certain special arithmetic transformations related to the ramanujan graphs of lubotzky phillips and sarnak the experiments indicate that the unfolded consecutive spacing distribution follows poisson statistics we provide a sharp estimate in that direction

correlations for pairs of closed geodesics in this article we consider natural counting problems for closed geodesics on negatively curved surfaces we present asymptotic estimates for pairs of closed geodesics the differences of whose lengths lie in a prescribed family of shrinking intervals related pair correlation problems have been studied in both quantum chaos and number theory

a schur method for pole assignment in this paper a new numerical algorithm for pole assignment of linear time invariant systems is presented the proposed algorithm uses reliable numerical techniques based on the schur form of the state matrix and on the use of orthogonal transformations only the bad eigenvalues of the system are modified by a sequential pole shifting procedure and the resulted gains in the feedback matrix are minimized

periodic orbits contribution to the 2 point correlation form factor for pseudo integrable systems using heuristic arguments based on the trace formulas we relate the 2 point correlation form factor k2 at small values of with sums over classical periodic orbits for typical examples of pseudo integrable systems the later sums have been explicitly calculated for the following models i plane billiards in the form of right triangles with one angle n and ii rectangular billiards with the aharonov bohm flux line in the first model using the properties of the veech structure it is shown that k2 0 n e n 3 n 2 where e n 0 for odd n e n 2 for even n not divisible by 3 and e n 6 for even n divisible by 3 for completeness we also recall informally the main features of the veech construction in the second model the answer depends on arithmetical properties of ratios of flux line coordinates to the corresponding sides of the rectangle when these ratios are non commensurable irrational numbers k2 0 1 3 where is the fractional part of the flux through the rectangle when and it is symmetric with respect to the line when the comparison of these results with numerical calculations of the form factor is discussed in detail the above values of k2 0 differ from all known examples of spectral statistics thus confirming analytically the peculiarities of statistical properties of the energy levels in pseudo integrable systems

on the periodic orbits of a strongly chaotic system abstract a point particle sliding freely on a two dimensional surface of constant negative curvature hadamard gutzwiller model exemplifies the simplest chaotic hamiltonian system exploiting the close connection between hyperbolic geometry and the group su 1 1 1 we construct an algorithm symboliv dynamics which generates the periodic orbits of the system for the simplest compact riemann surface having as its fundamental group the octagon group we present an enumeration of more than 206 million periodic orbits for the length of the n th primitive periodic orbit we find a simple expression in terms of algebraic numbers of the form m 2 n m n n are governed by a particular beatty sequence which reveals a strange arithmetical structure of chaos knowledge of the length spectrum is crucial for quantization via the selberg trace formula periodic orbit theory which in turn is expected to unravel the mystery of quantum chaos