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

a system of one dimensional balls with gravity we introduce a hamiltonian system with many degrees of freedom for which the nonvanishing of some lyapunov exponents almost everywhere can be established analytically

on correlations of cm type maass waveforms under the horocyclic flow maass waveforms of cm type are a special kind of eigenfunction of the hyperbolic laplacian whose defining components namely eigenvalue and fourier coefficients are given by simple formulae involving algebraic integers chosen from a suitable number field k in this paper we report on some computer experiments aimed at ascertaining the extent to which the autocorrelation behaviour of cm forms agrees with that of mock i e random waveforms in the limit of high energy our results suggest that no significant differences are seen

universal fluctuations of zeros of chaotic wavefunctions wavefunctions of one and two dimensional quantum systems can be parametrized by a finite number of zeros lying in phase space we study correlations of these zeros for fully chaotic systems in terms of a statistical model based on random polynomials excellent agreement is found for the two point correlation function and nearest neighbour spacing distribution of this model and the results obtained for wavefunctions of dynamical systems we conjecture that these correlation functions are valid for any chaotic system after rescaling the phase space distances unfolding some consequences for the distribution of zeros due to time reversal symmetry are also discussed

general quantum surface of section method a new method for exact quantization of general bound hamiltonian systems is presented it is the quantum analogue of the classical poincare surface of section sos reduction of classical dynamics the quantum poincare mapping is shown to be the product of the two generalized non unitary but compact on shell scattering operators of the two scattering hamiltonians which are obtained from the original bound one by cutting the f dimensional configuration space cs the along the f 1 dimensional configurational sos and attaching the flat quasi one dimensional waveguides instead the quantum poincare mapping has fixed points at the eigenenergies of the original bound hamiltonian the energy dependent quantum propagator e h 1 can be decomposed in terms of the four energy dependent propagators which propagate from and or to cs to and or from configurational sos which may generally be composed of many disconnected parts i show that in the semiclassical limit h cross to 0 the quantum poincare mapping converges to the bogomolny propagator and explain how the higher order semiclassical corrections can be obtained systematically

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

separating the regular and irregular energy levels and their statistics in a hamiltonian system with mixed classical dynamics we look at the high lying eigenstates from the 10001 to the 13000 in the robnik billiard defined as a quadratic conformal map of the unit disk with the shape parameter lambda 0 15 all the 3000 eigenstates have been numerically calculated and examined in the configuration space and in the phase space which in comparison with the classical phase space enabled a clear cut classification of energy levels into regular and irregular this is the first successful separation of energy levels based on purely dynamical rather than special geometrical symmetry properties we calculate the fractional measure of regular levels as rho 1 0 365 or 0 01 which is in remarkable agreement with the classical estimate rho 1 0 360 or 0 001 this finding confirms the percival s 1973 classification scheme the assumption in berry robnik 1984 theory and the rigorous result by lazutkin 1981 1991 the regular levels obey the poissonian statistics quite well whereas the irregular sequence exhibits the fractional power law level repulsion and globally brody like statistics with beta 0 286 or 0 001 this is due to the strong localization of irregular eigenstates in the classically chaotic regions therefore in the entire spectrum we see that the berry robnik regime is not yet fully established so that the level spacing distribution is correctly captured by the berry robnik brody distribution

new approach to the statistical properties of energy levels the joint distribution of energy eigenvalues of a hamiltonian is derived by means of the usual statistical laws of classical many body systems it makes a transition from the poisson type to the gaussian type depending on the value of a single parameter characteristic of the hamiltonian

quantum scars of classical closed orbits in phase space the way in which quantum eigenstates are influenced by the closed orbits of a chaotic classical system is analysed in phase space x q p through the spectral wigner function w x e this is a sum over wigner functions of eigenstates within a range of energy e in the classical limit w is concentrated on the energy surface and smoothly distributed over it closed orbits provide oscillatory corrections scars for which explicit semiclassical formulae are calculated each scar is a fringe pattern decorating the orbit as x moves off the energy surface the fringes form an airy pattern with spacing of order h as x moves off the closed orbit the fringes form a complex gaussian with spacing h

dyson s coulomb gas on a circle and intermediate eigenvalue statistics dyson s two dimensional coulomb gas on the unit circle with inverse temperature beta is investigated with the help of a metropolis algorithm theoretical predictions for energy and specific heat are verified the connection with the theory of ensembles of random unitary matrices with orthogonal unitary or symplectic symmetry corresponding to beta 1 2 4 is investigated in detail new approximation formulae for the spacing distribution and the so called delta statistics are proposed and found to be useful not only in the three cases mentioned but also for all beta between 0 and 4

distribution and fluctuation properties of transition probabilities in a system between integrability and chaos the authors study the statistical properties of transition probabilities or generalized intensities equal to the squared matrix elements of operators having a classical limit they first generalize the shnirelman theorem to include semiclassically integrable states and use this to generalize the feingold peres formula for the average value of generalized intensities they perform an unfolding procedure to separate the smooth mean part of the intensities as a function of frequency from its fluctuating part by applying the generalized feingold peres formula this formula relates the mean value of squared matrix elements to the power spectrum of the given observable over classical trajectories their approach is illustrated numerically by analysing the dipole transition probabilities in a family of billiards between integrability and chaos the average values of the intensities as a function of frequency are excellently described by the generalized feingold peres formula

a scattering approach to the quantization of hamiltonians in two dimensions application to the wedge billiard we extend the scattering approach to quantize general hamiltonian systems in two dimensions and demonstrate it for the wedge billiard the resulting energy levels are given by the zeros of a secular equation the corresponding wavefunctions may also be constructed we verify analytically that this quantization condition is exact in the case of the integrable wedges this method has been applied successfully to computing numerically a few thousand levels for various values of the angle using these data we can check the gutzwiller trace formula and other semiclassical relations involving the scattering matrix and periodic orbits for which we find excellent agreement

poincare map for scattering states for scattering states in classical mechanics a map is constructed which is an analogue to the poincare map for bound state trajectories iteration of this map indicates which parts of the phase space are filled by unstable trajectories that are sensitive to small changes in the initial conditions and which regions are filled by stable trajectories examples for this map are given by numerical calculations for scattering off a simple two dimensional model potential

nearest neighbour spacing distribution of energy levels in the region between integrability and chaos we derive an expression for the level spacing distribution for a quantal system displaying both regular and chaotic classical motion assuming that the level repulsion function for the mixed system is obtained by averaging the corresponding functions for the regular and chaotic regimes with weights given by their fractional phase space volumes the liouville measures of the regular region obtained from comparing this expression with the level spacing distributions of a hydrogen atom in a uniform magnetic field agree with the corresponding values obtained in the classical mechanical analysis of this system

classical dynamics of a family of billiards with analytic boundaries the classical dynamics of a billiard which is a quadratic conformal image of the unit disc is investigated the author gives the stability analysis of major periodic orbits present the poincare maps demonstrate the mixing properties by following the evolution of a small element in phase space show the existence of homoclinic points and calculate the lyapunov exponent and the kolmogorov entropy h it turns out that the system becomes strongly chaotic positive h for sufficiently large deformations of the unit disc the system shows a generic stochastic transition the computations suggest that the system is mixing if the boundary is not convex

cluster aggregation by shortest path travel the authors have studied a new growth process of cluster aggregation in this process particles are released randomly from a hypersphere enclosing the cluster and stick to those perimeter sites of the growing cluster which are closest at shortest distance to the releasing sites they find d 1 62 or 0 09 for d 2 and d 2 02 or 0 13 for d 3 for the fractal dimensions d of such aggregates in euclidean dimension d using hentschel s mean field theory for generalised cluster aggregations they argue that these aggregates should have the least fractal dimension because of maximum screening present in this model among all the aggregation models in which particles coming from outside form the aggregates the mean field values of d predicted by the theory are shown to be comparable with observation

non generic spectral statistics in the quantized stadium billiard we consider the effect of a continuous family of neutral bouncing ball orbits on the energy spectrum of the quantized stadium billiard using a semiclassical approximation we derive analytic expressions for standard two point spectral measures the corrections due to the bouncing ball orbits account for some of the non generic features observed in the analysis of the spectrum of a stadium cavity which was recently measured once the bouncing ball contributions are subtracted the spectrum is shown to be well reproduced by the semi classical trace formula based on unstable periodic orbits we also study special patterns in the spectrum which are due to other non generic features such as edge effects and whispering gallery trajectories

sensitivity of spin glass order to temperature changes by a loop expansion around parisi s mean field theory for an ising spin glass it is shown that the overlap of the magnetization patterns belonging to two different temperatures t and t vanishes to any order si t si t 0 while the correlation overlap sisj t sisj t calculated to first loop order and for technical reasons for dimensions d 8 only is found to decay exponentially with a characteristic length approximately t t 1

spectral statistics of a system with sharply divided phase space we construct a family of non kolmogorov arnold moser non kam piecewise linear continuous 2d area preserving maps which have sharply divided phase space with regions of regular elliptic and chaotic hyperbolic motion for such systems the shape of the islands of regular motion is either a solid ellipse or a solid polygon depending on the ir rationality of the frequency and thus the total area of the regular region of phase space can be computed analytically or at least rigorously estimated from below we analyse the spectral statistics for a few examples of the quantization of our maps and show that they provide a convenient playground for testing and confirming the key assumptions required for the validity of berry robnik formulae

berry robnik level statistics in a smooth billiard system the berry robnik level spacing distribution is demonstrated clearly in a generic quantized plane billiard for the first time however this ultimate semiclassical distribution is found to only be valid for extremely small semiclassical parameters effective planck s constant where the assumption of statistical independence of regular and irregular levels is achieved for sufficiently larger semiclassical parameters we find fractional power law level repulsion with a phenomenological brody distribution providing an adequate global fit

bogomolny s semiclassical transfer operator for rotationally invariant integrable systems the transfer operator due to bogomolny provides a convenient method for obtaining a semiclassical approximation to the energy eigenvalues of a quantum system no matter what the nature of the analogous classical system in this paper the method is applied to integrable systems which are rotationally invariant in two and three dimensions in two dimensions the transfer operator is expanded in a fourier series in the angle variable while in three dimensions it is expanded in spherical harmonics in both cases when the fourier coefficients are evaluated using the stationary phase approximation we arrive at the einstein brillouin keller quantization conditions the associated maslov indices are shown to agree with the results calculated by well known simple rules the theory is applied to several rotationally invariant systems including the hydrogen atom and the isotropic harmonic oscillator in two and three dimensions the circle billiard a billiard inside a spherical cavity and a harmonic potential with a singular magnetic flux line

parametric statistics of zeros of husimi representations of quantum chaotic eigenstates and random polynomials local parametric statistics of zeros of husimi representations of quantum eigenstates are introduced it is conjectured that for a classically fully chaotic system one should use the model of parametric statistics of complex roots of gaussian random polynomials which is exactly solvable as demonstrated below for example the velocities derivatives of zeros of husimi function with respect to an external parameter are predicted to obey a universal non maxwellian distribution where is the mean square velocity the conjecture is demonstrated numerically in a generic chaotic system with two degrees of freedom dynamical formulation of the zero flow in terms of an integrable many body dynamical system is given as well

a harnack inequality approach to the regularity of free boundaries part ii flat free boundaries are lipschitz soit u une solution faible d un probleme aux limites libres du type u 0 sur u 0 et sur u 0 et soit le long de la frontiere libre f la relation u v g u x est satisfaite au sens faible alors si f est proche du graphe d une fonction de lipschitz dans la boule unite b 1 de r n f est le graphe d une fonction de lipschitz dans b 1 2

the structure of two dimensional witten sander aggregates a relatively large number 102 103 of reasonably large about 105 particles or sites two dimensional witten sander aggregates are simulated using new algorithms this enables the reduction of statistical errors by an order of magnitude and provides an opportunity to determine some new quantities describing in more detail the structure of witten sander clusters a quantitate description of the distortion of large lattice based witten sander clusters into a diamond like shape is provided the orientational correlation in the bond directions for off lattice witten sander clusters is also measured this quantity has a power law dependence on distance with an exponent of about 0 37

a note on the level spacings distribution of the hamiltonians in the transition region between integrability and chaos it is well known that there is no strict universality of the spectral fluctuations of quantum hamiltonians whose classical counterparts undergo the transition from integrability to complete chaos the author discusses the level spacings distribution p s and explains why the semiclassical formulae of berry and robnik cannot be correct for small s there is no global universality of p s for nearly integrable systems but the approach to the integrability as the perturbation parameter in goes to zero can be universal this is reflected in the fact that the slope of dp ds at s 0 for small in is universally inversely proportional to in the author gives two models in terms of two dimensional random matrices one of them being based on maximum entropy considerations the author also points out the connection to the statistics of zeros of random functions and discusses the numerical evidence

regularity of the free boundary in parabolic phase transition problems in this paper we start the study of the regularity properties of the free boundary for parabolic two phase free boundary problems may be the best known example of a parabolic two phase free boundary problem is the stefan problem a simplified model describing the melting or solidification of a material with a solid liquid interphase the concept of solution can be stated in several ways classical solution weak so lution on divergence form or viscosity solution and as usual one would like to prove that the weak solutions that may be constructed are in fact as smooth and classical as possible locally a classical solution of the stefan problem may be described as following on the unit cylinder q1 b1 1 1 we have two complementary domains and qi separated by a smooth surface s oi2 nq1 in fl and qi we have two smooth solutions ul and u2 of the heat equations

correlations of chaotic eigenfunctions a semiclassical analysis we derive a semiclassical expression for an energy smoothed autocorrelation function defined on a group of eigenstates of the schr odinger equation the system we consider is an energy conserved hamiltonian system possessing time invariant symmetry the energy smoothed autocorrelation function is expressed as a sum of three terms the first one is analogous to berry s conjecture which is a bessel function of the zeroth order the second and the third terms are trace formulae made from special trajectories the second term is found to be direction dependent in the case of spacing averaging which agrees qualitatively with previous numerical observations in high lying eigenstates of a chaotic billiard

in out decomposition of boundary integral equations we propose a reformulation of the boundary integral equations for the helmholtz equation in a domain in terms of incoming and outgoing boundary waves we obtain transfer operator descriptions which are exact and thus incorporate features such as diffraction and evanescent coupling these effects are absent in the well known semiclassical transfer operators in the sense of bogomolny it has long been established that transfer operators are equivalent to the boundary integral approach within semiclassical approximation exact treatments have been restricted to specific boundary conditions such as dirichlet or neumann the approach we propose is independent of the boundary conditions and in fact allows one to decouple entirely the problem of propagating waves across the interior from the problem of reflecting waves at the boundary as an application we show how the decomposition may be used to calculate goos haenchen shifts of ray dynamics in billiards with variable boundary conditions and for dielectric cavities

failure of semiclassical methods to predict individual energy levels the authors argue that semiclassical methods quite generally cannot predict the individual energy levels not even in the semiclassical limit of small but finite and when the number of energy levels goes to infinity by this they mean that the average relative error of the semiclassical eigenvalues in units of the mean level spacing typically increases indefinitely as the energy goes to infinity or is at least bounded from below this they show for the case of the integrable circular billiard and the one dimensional potential uo cos2 alpha x by comparing the torus quantized semiclassical eigenenergies with the exact results since all the various semiclassical methods such as gutzwiller s and bogomolny s are reduced to the torus quantization in integrable cases they believe that their conclusion is generally valid they have theoretical arguments and strong numerical evidence for the case of the circular billiard that nevertheless the statistical properties of the exact energy spectra are correctly reproduced by the semiclassical approximations it is numerically found that the energy level spacing distribution and the spectral rigidity for the exact spectrum and for the semiclassical spectrum are in excellent agreement even for finite spectra where they both deviate from the limiting poissonian behaviour so they suggest that the nonuniversal approach to the limiting energy level statistics is also correctly described by the semiclassical theory they discuss the validity of semiclassical methods in the light of their negative and positive findings in addition they find the surprising result for the previously mentioned special cases that the error distribution of the semiclassical approximation is stationary i e it is independent of the energy

survey of the eigenfunctions of a billiard system between integrability and chaos we study numerically the eigenfunctions and their wigner phase space distributions of the two dimensional billiard system defined by the quadratic conformal image of the unit disk as introduced by robnik 1983 this system is a generic kam system and displays a transition from integrability to almost ergodicity as the billiard shape changes we clearly identify two classes of states the regular ones associated with integrable regions and the irregular states supported on classically chaotic regions whilst the mixed type states were not found in support of percival s conjecture 1973 we confirm the existence of extremely intense scars in the classically chaotic regions and demonstrate their association with classical periodic orbits three classes of scars are revealed one orbit scars many orbit one family scars of statistically similar orbits in the homoclinic neighbourhood and many orbit many family scars we argue that it is impossible to find an a priori semiclassical theory of individual eigenstates but do not deny the usefulness of general semiclassical arguments in analysing the collective and statistical properties of eigenstates

numerical demonstration of the semiclassical matrix element probability distribution between integrability and chaos in this letter i report on the first successful verification of the semiclassical matrix element probability distribution for the hamiltonian systems between integrability and chaos as for all the other statistical properties of quantum dynamical systems e g level spacing distribution phase space localization of eigenstates etc the semiclassical limit was found to converge extremely slowly so a rather abstract dynamical system was used namely the standard map on a torus in order to clearly demonstrate the semiclassical regime

amplitude distribution of eigenfunctions in mixed systems we study the amplitude distribution of irregular eigenfunctions in systems with mixed classical phase space for an appropriately restricted random wave model a theoretical prediction for the amplitude distribution is derived and a good agreement with numerical computations for the family of limacon billiards is found the natural extension of our result to more general systems e g with a potential is also discussed

exterior interior duality for discrete graphs the exterior interior duality expresses a deep connection between the laplace spectrum in bounded and connected domains in and the scattering matrices in the exterior of the domains here this link is extended to the study of the spectrum of the discrete laplacian on finite graphs for this purpose two methods are devised for associating scattering matrices to the graphs the exterior interior duality is derived for both methods

quantum section method for the soft stadium abstract the soft stadium is defined by a monomial potential with exponent as a parameter such that corresponds to the billiard the practical use of the quantum section method depends only on the partial separability of the system on both sides of the section which holds for all s in particular for 1 0 the system becomes globally separable allowing for a general test of the method for various values of the parameter we also tested the use of the asymptotic wkb type approximation in the construction of green s functions and asymptotic overlap integrals to obtain higher energy eigenvalues we show these approximations to be reliable

statistical properties of eigenstate amplitudes in complex quantum systems we study the eigenstates of quantum systems with large hilbert spaces via their distribution of wave function amplitudes in a real space basis for single particle quantum billiards these real space amplitudes are known to have gaussian distribution for chaotic systems in this work we formulate and address the corresponding question for many body lattice quantum systems for integrable many body systems we examine the deviation from gaussianity and provide evidence that the distribution generically tends toward power law behavior in the limit of large sizes we relate the deviation from gaussianity to the entanglement content of many body eigenstates for integrable billiards we find several cases where the distribution has power law tails

two phase free boundary problems for parabolic operators smoothness of the front we continue to develop the regularity theory of general two phase free boundary problems for parabolic operators in a 2010 paper we establish the optimal lipschitz regularity of a viscosity solution under the assumptions that the free boundary is locally a flat lipschitz graph and a nondegeneracy condition holds here on one side we improve this result by removing the nondegeneracy assumption on the other side we prove the smoothness of the front the proof relies in a crucial way on a local stability result stating that for a certain class of operators under small perturbations of the coefficients flat free boundaries remain close and flat 2013 wiley periodicals inc

statistical analysis of scars in stadium billiard in this paper by using our improved plane wave decomposition method we study the scars in the eigenfunctions of the stadium billiard from a very low state to as high as about the 1 millionth state in the systematic searching for scars of various types we have used the approximate criterion based on the quantization of the classical action along the unstable periodic orbit supporting the scar we have analysed the profile of the integrated probability density along the orbit we found that the maximal integrated intensity of different types of scars scales in different way with the which confirms qualitatively and quantitatively the existing theories of scars such as that of bogomolny and robnik

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

statistics of energy levels without time reversal symmetry aharonov bohm chaotic billiards a planar domain d contains a single line of magnetic flux phi switching on phi breaks time reversal symmetry t for quantal particles with charge q moving in d whilst preserving the geometry of classical billiard trajectories bouncing off the boundary delta d if delta d is such that these classical trajectories are chaotic the authors predict that t breaking will cause the local statistics of quantal energy levels to change their universality class from that of the gaussian orthogonal ensemble goe of random matrix theory to that of the gaussian unitary ensemble gue in the semiclassical limit this transition is abrupt for statistics involving the first n levels gue behaviour requires that the quantum flux alpha identical to q phi h 0 13n 14 the special flux alpha 1 2 corresponds to false t breaking and for this case goe statistics are predicted these predictions are confirmed by numerical computation of spectral statistics for a classically chaotic billiard without symmetry for which delta d is a cubic conformal image of the unit disc

random matrix theory models of electric grid topology abstract the random matrix theory is useful in the study of large systems such as electric grids these transmission systems can be modeled as complex networks with high voltage lines the edges that connect nodes representing power plants and substations we draw upon established literature of complex systems theory and introduce methods from nuclear and statistical physics to identify new characteristics of these networks we show that most grids can be characterized by the gaussian orthogonal ensemble an indicator of chaos in many complex systems under certain circumstances however grids may be described by poisson statistics an indicator of regularity we use the random matrix formalism to describe the interconnection of multiple grids and construct a simple model of a distributed grid

regularity of the free boundary for a bernoulli type parabolic problem with variable coefficients abstract in this work the regularity of the free boundary for a class of parabolic free boundary problems with variable coefficients is studied under the main hypothesis that the free boundary is lipschitz the main result is that the free boundary is in fact c 1 surfaces thereby implying that the viscosity solution is in fact classical in this case

two phase problems for linear elliptic operators with variable coefficients lipschitz free boundaries are c 1 γ we prove c 1 regularity of lipschitz free boundaries of two phase problems for linear elliptic operators with holder continuous coefficients

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

sensitivity of the eigenfunctions and the level curvature distribution in quantum billiards in searching for the manifestations of the sensitivity of the eigenfunctions in quantum billiards with dirichlet boundary conditions with respect to the boundary data the normal derivative we have performed instead various numerical tests for the robnik billiard a quadratic conformal map of the unit disk for 600 shape parameter values where we have looked at the sensitivity of the energy levels with respect to the shape parameter we show the energy level flow diagrams for three stretches of fifty consecutive odd eigenstates each with index 1000 to 2000 in particular we have calculated the unfolded and normalized level curvature distribution and found that it continuously changes from a delta distribution for the integrable case circle to a broad distribution in the classically ergodic regime for some shape parameters the agreement with the goe von oppen formula is very good whereas we have also cases where the deviation from goe is significant and of physical origin in the intermediate case of mixed classical dynamics we have a semiclassical formula in the spirit of the berry robnik 1984 surmise here the agreement with theory is not good partially due to the localization phenomena which are expected to disappear in the semiclassical limit we stress that even for classically ergodic systems there is no global universality for the curvature distribution not even in the semiclassical limit

spectral fluctuation properties of hamiltonian systems the transition region between order and chaos the authors study numerically the classical dynamical behaviour and the spectral fluctuation properties of a class of hamiltonian systems with two degrees of freedom the quantum mechanical properties of these systems are monotonic but non universal functions of the fraction of classical phase space filled by chaotic trajectories it is found that the observed spectral fluctuation measures can be reproduced by a random matrix model which depends on one parameter only

quantum phase space structures in classically mixed systems a scattering approach the structure of quantum wavefunctions is discussed for a billiard system with mixed classical dynamics using a scattering formalism we introduce a husimi like distribution for s matrix eigenvectors at arbitrary wavenumbers k not necessarily eigenenergies of the system this constitutes probability density plots on the poincare surface of section at any k many eigenvectors are structured by classical objects such as invariant tori the intermediate layer between regular and chaotic motion and unstable periodic orbits scars we find that eigenvector structure is stable over significant energy ranges while the associated eigenphases increase linearly with k at a rate determined by the underlying classical objects additionally we observe that in the presence of time reversal symmetry eigenvectors scarred by non self retracing periodic orbits form doublets which are closely spaced in eigenphase our results provide a new perspective on the quantization of structured states quantization is determined by the linearly increasing eigenphases and occurs whenever an eigenphase is equal to while structure is encoded in the eigenvectors it is present at all energies and varies only slowly this allows one to predict many quantized eigenfunctions over an entire energy range by studying the eigenparameters of at a single intermediate energy and identifying the underlying classical objects

aggregation by attractive particle cluster interaction a quasideterministic irreversible growth model is studied where aggregates are generated by attractive cluster particle forces proportional ro r alpha it is shown that fractal dimensions and growth site probability measures of the resulting fractal structures strongly depend on the parameter alpha comparison with experimental results indicates that this model represents a further step towards realistic simulations of growth processes

a free boundary problem for the heat equation arising in flame propagation we introduce a new free boundary problem for the heat equation of interest in combustion theory it is obtained in the description of laminar flames as an asymptotic limit for high activation energy the problem asks for the determination of a domain in space time q c rn x 0 t and a function u x t 0 defined in q such that ut au in q u takes certain initial conditions u x 0 uo x for x e qo oqfn t 0 and two conditions are satisfied on the free boundary r a2 n t 0 u o and u1 1 where u1 denotes the derivative of u along the spatial exterior normal to r we approximate this problem by means of a certain regularization on the boundary and prove the existence of a weak solution under suitable assumptions on the initial data

level repulsion near integrability a random matrix analogy using the analogy between the statistics of the levels of quantum hamiltonians and the eigenvalues of random matrices the authors use an appropriate choice of the latter in order to study the transition region near integrability they show that the nearest neighbour spacing distribution is linear for small spacings while the inverse of its slope is proportional to the amplitude of the integrability destroying perturbation

quantum surface of section method demonstration of semiclassical berry robnik energy level spacing distribution in a generic two dimensional hamiltonian system the recently developed quantum surface of section method is applied to a search for extremely high lying energy levels in a simple but generic hamiltonian system between integrability and chaos namely the semiseparable two dimensional oscillator using the stretch of 13445 consecutive levels with the sequential number around 1 8 107 eighteen million we have clearly demonstrated the validity of the semiclassical berry robnik level spacing distribution while at 1000 times smaller sequential quantum numbers we find the very persistent quasi universal phenomenon of power law level repulsion which is globally very well described by the phenomenological brody distribution this is the first statistically significant numerical confirmation of the berry robnik surmise in an autonomous two dimensional hamiltonian system

numerical accuracy of bogomolny s semiclassical quantization scheme in quantum billiards we use the semiclassical quantization scheme of bogomolny to calculate eigenvalues of the limacon quantum billiard corresponding to a conformal map of the circle billiard we use the entire billiard boundary as the chosen surface of section and use a finite approximation for the transfer operator in coordinate space computation of the eigenvalues of this matrix combined with a quantization condition determines a set of semiclassical eigenvalues which are compared with those obtained by solving the schrodinger equation the classical dynamics of this billiard system undergoes a smooth transition from integrable circle to completely chaotic motion thus providing a test of bogomolny s semiclassical method in coordinate space in terms of the morphology of the wavefunction we analyse the results for billiards which exhibit both soft and hard chaos

logarithmic terms in the spectral statistics of band random matrices the authors present a detailed analysis of the nearest neighbour spacing distribution nnsd for random matrices on which the non zero elements are confined to a band around the diagonal the vanishing of the remaining elements induces non analytic logarithmic behaviour for the nnsd at very small spacings a conjective for the precise behaviour of these logarithmic terms is presented and supported by extensive numerical computations

a free energy bound for the hopfield model the author gives a simple upper and lower bound on the free energy density of the hopfield model of size n with p stored patterns in the limit where n and p tend to infinity with p n to alpha 1 the two bounds coincide for alpha 0

spectral duality for planar billiards for a bounded open domain with connected complement inr2 and piecewise smooth boundary we consider the dirichlet laplacian on and the s matrix on the complement c we show that the on shell s matricessk have eigenvalues converging to 1 ask k0 exactly when has an eigenvalue at energyk02 this includes multiplicities and proves a weak form of transparency atk k0 we also show that stronger forms of transparency such assk0 having an eigenvalue 1 are not expected to hold in general

exact quantum surface of section method this letter presents exact surface of section reduction of quantum mechanics the main theoretical result is a decomposition of the resolvent of the autonomous bound hamiltonian h e h 1 in terms of the four propagators which propagate from and or to hilbert space over the configuration space cs to and or from hilbert space over the configurational surface of section csos which has one dimension less than cs the exact energy quantization condition can be expressed solely in terms of the csos csos propagator all these newly defined energy dependent cs csos cs csos propagators are expressed in terms of the resolvent of the related scattering hamiltonian

study of regular and irregular states in generic systems in this work we present the results of a numerical and semiclassical analysis of high lying states in a hamiltonian system whose classical mechanics is of a generic mixed type where the energy surface is split into regions of regular and chaotic motion as predicted by the principle of uniform semiclassical condensation when the effective tends to zero each state can be classified as regular or irregular we were able to semiclassically reproduce individual regular states by the einstein brillouin keller torus quantization for which we devise a new approach while for the irregular ones we found the semiclassical prediction of their autocorrelation function in a good agreement with numerics we also looked at the low lying states to get a better understanding of the onset of semiclassical behaviour

statistics of wave functions in mesoscopic systems we review the results of a recent study of fluctuations of wave functions in confined chaotic systems the fluctuations can be due to a random potential or be a consequence of a chaotic scattering by the walls the entire distribution function of the local amplitudes of the wave functions f 1 and the joint two point distribution are calculated in various situations the computation is performed using the supersymmetry technique and employs the studies of a reduced version of the non linear supersymmetric model developed especially for investigating the properties of a single eigenstate in a discrete spectrum of a chaotic quantum system for not very large amplitudes the complete description can be achieved using the zero dimensional approximation of the model the distribution function calculated in the limit of various symmetry classes shows the universal behavior known as the porter thomas statistics and fluctuations at distant points do not correlate in the crossover regime between the ensembles the distribution of local amplitudes shows a somewhat more sophisticated behavior the fluctuations in this case are correlated over distances exceeding the mean free path for large amplitudes generated by the states the most affected by the localization we call them prelocalized the zero dimensional approximation is no longer valid instead the statistics of their wave functions is determined by nontrivial vacua of the reduced model which is quite similar to the liouville model known in conformal field theory in particular the vacuum state of the reduced model obeys the liouville equation which indicates that in two dimensions the prelocalized states have nearly critical properties we prove their multifractality and power law statistically averaged envelope r 2 r 2 at the intermediate range of distances below the localization length with a spectrum of exponents 1 as well as obtain a logarithmically normal tail of the distribution functionf 1 we also find an evidence of prelocalized states in quasi one dimensional wires with the length shorter than the localization length their statistically averaged envelope has power law asymptotics x 2 x 2 and the tail of the distribution function is similar to that describing localized states in the infinite wires

caloric functions in lipschitz domains and the regularity of solutions to phase transition problems in some domain d c rn l that vanishes locally on some distinguished part of od the term refers to the fact that in some sense along that part of od u resembles the fundamental solution of the heat equation in d this is particularly the case when the domain under consideration d is the intersection of some n 1 dimensional cube q with q one side of a lipschitz graph i e q fx f x t that is d q n q and the distinguished part of od is precisely q n q this is the main first area of study of this paper relying on work of fabes garofalo salsa fgs on backward harnack type inequalities for such domains lipschitz regularity in time versus lipschitz regularity in space is not of course the natural homogeneity balance for the study of parabolic equations but it is so for the study of phase transition relations of the form

smoothed wave functions of chaotic quantum systems abstract it is shown that wave functions of quantum systems as 0 have an extra density near unstable periodic trajectories of the classical problem the averaged wave function square is represented as the sum over a finite number of periodic trajectories the contribution of each trajectory is expressed through the elements of the monodromy matrix of the trajectory the results are compared with the numerical calculations of the wave functions for the stadium billiard

autocorrelation function of eigenstates in chaotic and mixed systems we study the autocorrelation function of different types of eigenfunctions in quantum mechanical systems with either chaotic or mixed classical limits we obtain an expansion of the autocorrelation function in terms of the correlation distance for localized states in billiards like bouncing ball modes or states living on tori a simple model using only classical input gives good agreement with the exact result in particular a prediction for irregular eigenfunctions in mixed systems is derived and tested for chaotic systems the expansion of the autocorrelation function can be used to test quantum ergodicity on different length scales

statistical properties of spectral fluctuations for a quantum system with infinitely many components extending the idea formulated in makino phys rev e 67 066205 2003 that is based on the berry robnik approach m v berry and m robnik j phys a 17 2413 1984 we investigate the statistical properties of a two point spectral correlation for a classically integrable quantum system the eigenenergy sequence of this system is regarded as a superposition of infinitely many independent components in the semiclassical limit we derive the level number variance lnv in the limit of infinitely many components and discuss its deviations from poisson statistics the slope of the limiting lnv is found to be larger than that of poisson statistics when the individual components have a certain accumulation this property agrees with the result from the semiclassical periodic orbit theory that is applied to a system with degenerate torus actions d biswas m azam and s v lawande phys rev a 43 5694 1991

walks trials and polymers with loops lattice models interpolating between free and self avoiding random walks are investigated generating functions are constructed for k tolerant walks various trail problems etc for trail problems the effective field theory describing the global behaviour is analysed in the vicinity of the upper critical dimension dc 4 their asymptotic large scale behaviours are the same as those of the self avoiding walk arguments are presented to support the same conclusion for much more general classes of walks including k tolerant walks one of the models exhibits a new tricritical point of order epsilon 1 2 if the fugacity for crossing is increased

statistical properties of high lying chaotic eigenstates we study the statistical properties of the high lying chaotic eigenstates 200000 and above which are deep within the semiclassical regime the system we are analysing is the billiard system inside the region defined by the quadratic complex conformal map of the unit disk as introduced by robnik 1983 we are using heller s method of plane wave decomposition of the numerical eigenfunctions and perform extensive statistical analysis with the following conclusions i the local average probability density is in excellent agreement with the microcanonical assumption and all statistical properties are also in excellent agreement with the gaussian random model ii the autocorrelation function is found to be strongly direction dependent and only after averaging over all directions agrees well with berry s 1977 prediction iii although the scars of unstable classical periodic orbits in such an ergodic regime are expected to exist so far we have not found any around the 200000th state other than a scar like feature resembling the whispering gallery modes

level spacing for band random matrices the authors examine the spacing distribution of the eigenvalues of tridiagonal real symmetric random matrices the elements of which are distributed according to a gaussian law they show explicitly that for 4 4 matrices the distribution at small spacings behaves as s log2 s they surmise that for n n matrices the behaviour is s logn 2s and they present numerical results which support this conjecture

energy level statistics and localization in sparsed banded random matrix ensemble the matrix elements of the quantum hamiltonian which corresponds to the classical kam system h h0 epsilon v where h0 is integrable in an energy ordered eigenbasis of h0 are considered their statistical properties are comprised in the definition of the sparsed banded random matrix ensemble sbrme in analysing the spectral statistics of this ensemble the authors find the power law level repulsion and the level spacing distribution which is well described by the brody distribution the numerical study of sbrme shows two interesting scaling laws i the connection between the scaling variable x alpha b3 2 alpha mean diagonal increment b bandwidth and the level repulsion parameter beta as deduced from the brody distribution and ii the connection between the same scaling variable x and an expression which contains two types of localization length entropy and geometric the universal aspects and the importance of sbrme for the hamiltonian systems in the transition region between integrability and chaos are discussed

on the discontinuities of the boundary in billiards abstract this work concerns billiards in domains of r 2 the set s of pre images of discontinuities of t the bouncing map induced by the flow is studied particularly those discontinuities due to points where the boundary is not c1 we prove that if the angle at the discontinuity is a divisor of there exists a power of t which is continuous around such a point and that under conditions of symmetry this power is locally differentiable it is assumed that continuity may often be extended to certain areas of phase space this property is used to show by means of numerical simulations of two billiards how s fashions the phase space in general if the angle is not a divisor of s separates the distinct areas of quasi interagrable motion one of the billiards is isomorphic to a mechanical model of three interacting particles on a line

nodal statistics on quantum graphs it has been suggested that the distribution of the suitably normalized number of zeros of laplacian eigenfunctions contains information about the geometry of the underlying domain we study this distribution more precisely the distribution of the nodal surplus for laplacian eigenfunctions of a metric graph the existence of the distribution is established along with its symmetry one consequence of the symmetry is that the graph s first betti number can be recovered as twice the average nodal surplus of its eigenfunctions furthermore for graphs with disjoint cycles it is proven that the distribution has a universal form it is binomial over the allowed range of values of the surplus to prove the latter result we introduce the notion of a local nodal surplus and study its symmetry and dependence properties establishing that the local nodal surpluses of disjoint cycles behave like independent bernoulli variables

spectral statistics in the quantized cardioid billiard the spectral statistics of the strongly chaotic cardioid billiard is studied the analysis is based on the first 11 000 quantal energy levels for both odd and even symmetry it is found that the level spacing distribution is in good agreement with the gaussian orthogonal ensemble distribution of random matrix theory in the cases of the number variance and rigidity we observe agreement with the random matrix model for short range correlations only whereas for long range correlations both statistics saturate in agreement with semiclassical expectations furthermore the conjecture that for classically chaotic systems the normalized mode fluctuations have a universal gaussian distribution with unit variance is tested and found to be in very good agreement for both symmetry classes by means of the gutzwiller trace formula the trace of the cosine modulated heat kernel is studied since the billiard boundary is focusing there are conjugate points that give rise to zeros instead of exclusively at gaussian peaks at the locations of the periodic orbits

comment on energy level statistics in the mixed regime we comment on the recent paper by abul magd 1996 29 1 concerning the energy level statistics in the mixed regime i e such having the mixed classical dynamics where regular and chaotic regions coexist in the phase space we point out that his basic assumption on the additive property of the level repulsion function r s conditional probability density in the sense of dividing it linearly into the regular and chaotic part in proportion to the classical fractional phase space volumes and is not justified since among other things it relies on the type of berry s ergodic assumption which however is only correct in a homogeneous ensemble of ergodic systems not in the neighbourhood of an integrable system thus his resulting distribution cannot be regarded as a theoretically well founded object we point out that the semiclassical limiting energy level spacing distribution must be of berry robnik 1984 type and explain what transitional behaviour of the brody type with fractional power law energy level repulsion we observe in the near semiclassical regime where effective is not yet small enough thus we refer to the derivation arguments and conclusions in our previous paper prosen and robnik 1994 26 8059 and explain again the behaviour in this double transition region

wavefunctions expectation values and scars on poincare sections a scattering approach using the scattering approach to quantization we define an appropriate representation of the billiard wavefunction on the scattering poincar section the expectation values of smooth operators in terms of these poincar section wavefunctions are expressed as sums over periodic orbits a special operator is used to define scars on the section and the relation to scars in configuration space is discussed and demonstrated numerically

intermediate statistics of the quasi energy spectrum and quantum localisation of classical chaos the statistical properties of the quasi energy spectrum in a simple quantum model are investigated for the case when the corresponding classical system is fully chaotic while quantum chaos is restricted by the localisation effects it is shown that the level spacing distribution depends effectively on some parameter which is the ratio of the dimension of the eigenfunctions mean localisation length to the total number of the quasi energy levels numerical data for a wide range of parameters of the system are given