statistics on diffeomorphisms via tangent space representations in this paper we present a linear setting for statistical analysis of shape and an optimization approach based on a recent derivation of a conservation of momentum law for the geodesics of diffeomorphic flow once a template is fixed the space of initial momentum becomes an appropriate space for studying shape via geodesic flow since the flow at any point along the geodesic is completely determined by the momentum at the origin through geodesic shooting equations the space of initial momentum provides a linear representation of the nonlinear diffeomorphic shape space in which linear statistical analysis can be applied specializing to the landmark matching problem of computational anatomy we derive an algorithm for solving the variational problem with respect to the initial momentum and demonstrate principal component analysis pca in this setting with three dimensional face and hippocampus databases

on level spacing distributions for 2d non normal gaussian random matrices the level spacing distribution of general non normal gaussian random 2d matrices is derived in particular tridiagonal matrices have no level repulsion and show a halfsided gaussian distribution general non normal matrices show strong level repulsion the repulsion exponent is 2 0log

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

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

geometric atlas modeling the cortex as an organized surface recent atlases of the cortical surface are based on a modelization of the cerebral cortex as a topological sphere this captures effectively its organization as a regular bidimensional sheet of layers parallel to the surface and with perpendicular cortical columns yet while in the vertical direction cortices are almost the same throughout phylia in the sense of its surface the cerebral cortex is one of the most variable and distinctive parts of the nervous system indeed gyri and sulci appear to have a crucial organizing role in an architectonic connectional and functional sense this organization is not explicitly captured by the surface model of the cortex we propose a geometric model of the cortical anatomy based on flat representations of principal sulci obtained from surface reconstructions of mri data and on neuroanatomical and theoretical considerations concerning the folding patterns of the cortex the cortex is modeled by a sphere where primary sulci are included as axes the arrangement of the axes is a simplification of the arrangement of principal sulci observed in flat stereographic representations of the whole cortical surface the position of secondary and tertiary sulci is then defined by a field of orientations parallel and orthogonal to the axes we consider the use of the geometric model as a synthetic reference cortex for addressing reconstructions of cortical surfaces we present a method which establishes a bijection between the geometric model and a cortical surface reconstruction by using the axes of the model as boundary conditions for a set of partial differential equations solved over both surfaces using the geometric model as atlas provides a natural parameterization of the cortical surface that unlike angular coordinates allows for a localization based on the surface distance to its main organizing landmarks and folding patterns

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

whole brain diffeomorphic metric mapping via integration of sulcal and gyral curves cortical surfaces and images this paper introduces a novel large deformation diffeomorphic metric mapping algorithm for whole brain registration where sulcal and gyral curves cortical surfaces and intensity images are simultaneously carried from one subject to another through a flow of diffeomorphisms to the best of our knowledge this is the first time that the diffeomorphic metric from one brain to another is derived in a shape space of intensity images and point sets such as curves and surfaces in a unified manner we describe the euler lagrange equation associated with this algorithm with respect to momentum a linear transformation of the velocity vector field of the diffeomorphic flow the numerical implementation for solving this variational problem which involves large scale kernel convolution in an irregular grid is made feasible by introducing a class of computationally friendly kernels we apply this algorithm to align magnetic resonance brain data our whole brain mapping results show that our algorithm outperforms the image based lddmm algorithm in terms of the mapping accuracy of gyral sulcal curves sulcal regions and cortical and subcortical segmentation moreover our algorithm provides better whole brain alignment than combined volumetric and surface registration postelnicu et al 2009 and hierarchical attribute matching mechanism for elastic registration hammer shen and davatzikos 2002 in terms of cortical and subcortical volume segmentation

large deformation diffeomorphic metric curve mapping we present a matching criterion for curves and integrate it into the large deformation diffeomorphic metric mapping lddmm scheme for computing an optimal transformation between two curves embedded in euclidean space d curves are first represented as vector valued measures which incorporate both location and the first order geometric structure of the curves then a hilbert space structure is imposed on the measures to build the norm for quantifying the closeness between two curves we describe a discretized version of this in which discrete sequences of points along the curve are represented by vector valued functionals this gives a convenient and practical way to define a matching functional for curves we derive and implement the curve matching in the large deformation framework and demonstrate mapping results of curves in 2 and 3 behaviors of the curve mapping are discussed using 2d curves the applications to shape classification is shown and experiments with 3d curves extracted from brain cortical surfaces are presented

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

hammer hierarchical attribute matching mechanism for elastic registration a new approach is presented for elastic registration of medical images and is applied to magnetic resonance images of the brain experimental results demonstrate very high accuracy in superposition of images from different subjects there are two major novelties in the proposed algorithm first it uses an attribute vector i e a set of geometric moment invariants gmis that are defined on each voxel in an image and are calculated from the tissue maps to reflect the underlying anatomy at different scales the attribute vector if rich enough can distinguish between different parts of an image which helps establish anatomical correspondences in the deformation procedure it also helps reduce local minima by reducing ambiguity in potential matches this is a fundamental deviation of our method referred to as the hierarchical attribute matching mechanism for elastic registration hammer from other volumetric deformation methods which are typically based on maximizing image similarity second in order to avoid being trapped by local minima i e suboptimal poor matches hammer uses a successive approximation of the energy function being optimized by lower dimensional smooth energy functions which are constructed to have significantly fewer local minima this is achieved by hierarchically selecting the driving features that have distinct attribute vectors thus drastically reducing ambiguity in finding correspondence a number of experiments demonstrate that the proposed algorithm results in accurate superposition of image data from individuals with significant anatomical differences

quantitative evaluation of lddmm freesurfer and caret for cortical surface mapping cortical surface mapping has been widely used to compensate for individual variability of cortical shape and topology in anatomical and functional studies while many surface mapping methods were proposed based on landmarks curves spherical or native cortical coordinates few studies have extensively and quantitatively evaluated surface mapping methods across different methodologies in this study we compared five cortical surface mapping algorithms including large deformation diffeomorphic metric mapping lddmm for curves lddmm curve for surfaces lddmm surface multi manifold lddmm mm lddmm freesurfer and caret using 40 mri scans and 10 simulated datasets we computed curve variation errors and surface alignment consistency for assessing the mapping accuracy of local cortical features e g gyral sulcal curves and sulcal regions and the curvature correlation for measuring the mapping accuracy in terms of overall cortical shape in addition the simulated datasets facilitated the investigation of mapping error distribution over the cortical surface when the mm lddmm freesurfer and caret mapping algorithms were applied our results revealed that the lddmm curve mm lddmm and caret approaches best aligned the local curve features with their own curves the mm lddmm approach was also found to be the best in aligning the local regions and cortical folding patterns e g curvature as compared to the other mapping approaches the simulation experiment showed that the mm lddmm mapping yielded less local and global deformation errors than the caret and freesurfer mappings

dynamic programming generation of boundaries of local coordinatized submanifolds in the neocortex application to the planum temporale dynamic programming is used to define boundaries of cortical submanifolds with focus on the planum temporale pt of the superior temporal gyrus stg which has been implicated in a variety of neuropsychiatric disorders to this end automated methods are used to generate the pt manifold from 10 high resolution mri subvolumes roi masks encompassing the stg a procedure to define the subvolume roi masks from original mri brain scans is developed bayesian segmentation is then used to segment the subvolumes into cerebrospinal fluid gray matter gm and white matter wm 3d isocontouring using the intensity value at which there is equal probability of gm and wm is used to reconstruct the triangulated graph representing the stg cortical surface enabling principal curvature at each point on the graph to be computed dynamic programming is used to delineate the pt manifold by tracking principal curves from the retro insular end of the heschl s gyrus hg to the stg along the posterior stg up to the start of the ramus and back to the retro insular end of the hg a coordinate system is then defined on the pt manifold the origin is defined by the retro insular end of the hg and the y axis passes through the point on the posterior stg where the ramus begins automated labeling of gm in the stg is robust with l 1 distances between bayesian and manual segmentation in the range 0 001 0 12 n 20 pt reconstruction is also robust with 90 of the vertices of the reconstructed pt within about 1 voxel n 20 from semiautomated contours finally the reliability index based on interrater intraclass correlation for the surface area derived from repeated reconstructions is 0 96 for the left pt and 0 94 for the right pt thus demonstrating the robustness of dynamic programming in defining a coordinate system on the pt it provides a method with potential significance in the study of neuropsychiatric disorders

spectral fluctuations of billiards with mixed dynamics from time series to superstatistics a statistical analysis of the eigenfrequencies of two sets of superconducting microwave billiards one with mushroomlike shape and the other from the family of the limacon billiards is presented these billiards have mixed regular chaotic dynamics but different structures in their classical phase spaces the spectrum of each billiard is represented as a time series where the level order plays the role of time two most important findings follow from the time series analysis first the spectra can be characterized by two distinct relaxation lengths this is a prerequisite for the validity of the superstatistical approach which is based on the folding of two distribution functions second the shape of the resulting probability density function of the so called superstatistical parameter is reasonably approximated by an inverse chi2 distribution this distribution is used to compute nearest neighbor spacing distributions and compare them with those of the resonance frequencies of billiards with mixed dynamics within the framework of superstatistics the obtained spacing distribution is found to present a good description of the experimental ones and is of the same or even better quality as a number of other spacing distributions including the one from berry and robnik however in contrast to other approaches toward a theoretical description of spectral properties of systems with mixed dynamics superstatistics also provides a description of properties of the eigenfunctions in terms of a superstatistical generalization of the porter thomas distribution indeed the inverse chi2 parameter distribution is found suitable for the analysis of experimental resonance strengths in the limacon billiards within the framework of superstatistics

the emerging discipline of computational functional anatomy computational functional anatomy cfa is the study of functional and physiological response variables in anatomical coordinates for this we focus on two things i the construction of bijections via diffeomorphisms between the coordinatized manifolds of human anatomy and ii the transfer group action and parallel transport of functional information into anatomical atlases via these bijections we review advances in the unification of the bijective comparison of anatomical submanifolds via point sets including points curves and surface triangulations as well as dense imagery we examine the transfer via these bijections of functional response variables into anatomical coordinates via group action on scalars and matrices in dti as well as parallel transport of metric information across multiple templates which preserves the inner product

consequences of flooding on spectral statistics we study spectral statistics in systems with a mixed phase space in which regions of regular and chaotic motion coexist increasing their density of states we observe a transition of the level spacing distribution p s from berry robnik to wigner statistics although the underlying classical phase space structure and the effective planck constant h eff remain unchanged this transition is induced by flooding i e the disappearance of regular states due to increasing regular to chaotic couplings we account for this effect by a flooding improved berry robnik distribution in which an effectively reduced size of the regular island enters to additionally describe power law level repulsion at small spacings we extend this prediction by explicitly considering the tunneling couplings between regular and chaotic states this results in a flooding and tunneling improved berry robnik distribution which is in excellent agreement with numerical data

dynamical localization in chaotic systems spectral statistics and localization measure in the kicked rotator as a paradigm for time dependent and time independent systems we study the kicked rotator in the classically fully chaotic regime using izrailev s n dimensional model for various n 4000 which in the limit n tends to the quantized kicked rotator we do treat not only the case k 5 as studied previously but also many different values of the classical kick parameter 5 k 35 and many different values of the quantum parameter ke 5 60 we describe the features of dynamical localization of chaotic eigenstates as a paradigm for other both time periodic and time independent autonomous fully chaotic or and mixed type hamilton systems we generalize the scaling variable l n to the case of anomalous diffusion in the classical phase space by deriving the localization length l for the case of generalized classical diffusion we greatly improve the accuracy and statistical significance of the numerical calculations giving rise to the following conclusions 1 the level spacing distribution of the eigenphases or quasienergies is very well described by the brody distribution systematically better than by other proposed models for various brody exponents br 2 we study the eigenfunctions of the floquet operator and characterize their localization properties using the information entropy measure which after normalization is given by loc in the interval 0 1 the level repulsion parameters br and loc are almost linearly related close to the identity line 3 we show the existence of a scaling law between loc and the relative localization length now including the regimes of anomalous diffusion the above findings are important also for chaotic eigenstates in time independent systems batistic and robnik j phys a math gen 43 215101 2010 arxiv 1302 7174 2013 where the brody distribution is confirmed to a very high degree of precision for dynamically localized chaotic eigenstates even in the mixed type systems after separation of regular and chaotic eigenstates

stickiness in hamiltonian systems from sharply divided to hierarchical phase space we investigate the dynamics of chaotic trajectories in simple yet physically important hamiltonian systems with nonhierarchical borders between regular and chaotic regions with positive measures we show that the stickiness to the border of the regular regions in systems with such a sharply divided phase space occurs through one parameter families of marginally unstable periodic orbits and is characterized by an exponent ensuremath gamma 2 for the asymptotic power law decay of the distribution of recurrence times generic perturbations lead to systems with hierarchical phase space where the stickiness is apparently enhanced due to the presence of infinitely many regular islands and cantori in this case we show that the distribution of recurrence times can be composed of a sum of exponentials or a sum of power laws depending on the relative contribution of the primary and secondary structures of the hierarchy numerical verification of our main results are provided for area preserving maps mushroom billiards and the newly defined magnetic mushroom billiards

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

distribution of the wave function inside chaotic partially open systems we demonstrate both theoretically and experimentally that the distribution of the wave function inside a partially open chaotic time reversal symmetric system displays significant deviations from the porter thomas distribution we give arguments which show that this distribution resembles the distribution which is expected to be found in closed chaotic systems with broken time reversal symmetry

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

multi manifold diffeomorphic metric mapping for aligning cortical hemispheric surfaces cortical surface based analysis has been widely used in anatomical and functional studies because it is geometrically appropriate for the cortex one of the main challenges in the cortical surface based analysis is to optimize the alignment of the cortical hemispheric surfaces across individuals in this paper we introduce a multi manifold large deformation diffeomorphic metric mapping mm lddmm algorithm that allows simultaneously carrying the cortical hemispheric surface and its sulcal curves from one to the other through a flow of diffeomorphisms we present an algorithm based on recent derivation of a law of momentum conservation for the geodesics of diffeomorphic flow once a template is fixed the space of initial momentum becomes an appropriate space for studying shape via geodesic flow since the flow at any point on curves and surfaces along the geodesic is completely determined by the momentum at the origin we solve for trajectories geodesics of the kinetic energy by computing its variation with respect to the initial momentum and by applying a gradient descent scheme the mm lddmm algorithm optimizes the initial momenta encoding the anatomical variation of each individual relative to a common coordinate system in a linear space which provides a natural scheme for shape deformation average and template or atlas generation we applied the mm lddmm algorithm for constructing the templates for the cortical surface and 14 sulcal curves of each hemisphere using a group of 40 subjects the estimated template shape reflects regions which are highly variable across these subjects compared with existing single manifold lddmm algorithms such as the lddmm curve mapping and the lddmm surface mapping the mm lddmm mapping provides better results in terms of surface to surface distances in five predefined regions

semiempirical theory of level spacing distribution beyond the berry robnik regime modeling the localization and the tunneling effects in this work we study the level spacing distribution in the classically mixed type quantum systems which are generic exhibiting regular motion on invariant tori for some initial conditions and chaotic motion for the complementary initial conditions in the asymptotic regime of the sufficiently deep semiclassical limit sufficiently small effective planck constant the berry and robnik 1984 j phys a math gen 17 2413 picture applies which is very well established we present a new quasi universal semiempirical theory of the level spacing distribution in a regime away from the berry robnik regime the near semiclassical limit by describing both the dynamical localization effects of chaotic eigenstates and the tunneling effects which couple regular and chaotic eigenstates the theory works extremely well in the 2d mixed type billiard system introduced by robnik 1983 j phys a math gen 16 3971 and is also tested in other systems mushroom billiard and prosen billiard

triangular sos models and cubic crystal shapes a solid on solid sos model in a field h conjugate to the orientation of the surface is exactly solved with the aid of pfaffians the free energy h directly gives the equilibrium shape of a finite crystal the phase diagram exhibits rough and smooth phases corresponding to rounded and flat portions of the crystal surface the solid on solid model undergoes transitions of the pokrovsky talapov type 1979 characterised by a specific heat exponent alpha 1 2 one special point of the phase diagram corresponds to the appearance of a facet via an alpha 0 transition height height correlations are derived along a special line in the phase diagram with the aid of the known equivalence of this sos model with an ising model several exponents can be translated from one model to the other this enables one to derive the topology of the phase diagram of the antiferromagnetic triangular ising model with first and second neighbour couplings in a field

dynamical localization of chaotic eigenstates in the mixed type systems spectral statistics in a billiard system after separation of regular and chaotic eigenstates we study the quantum mechanics of a billiard robnik 1983 j phys a math gen 16 3971 in the regime of mixed type classical phase space the shape parameter 0 15 at very high lying eigenstates starting at about 1 000 000th eigenstate and including the consecutive 587654 eigenstates by calculating the normalized poincare husimi functions of the eigenstates and comparing them with the classical phase space structure we introduce the overlap criterion which enables us to separate with great accuracy and reliability the regular and chaotic eigenstates and the corresponding energies the chaotic eigenstates appear all to be dynamically localized meaning that they do not uniformly occupy the entire available chaotic classical phase space component but are localized on a proper subset we find with unprecedented precision and statistical significance that the level spacing distributions of the regular levels obey the poisson statistics and the chaotic ones obey the brody statistics as anticipated in a recent paper by batistic and robnik 2010 j phys a math theor 43 215101 where the entire spectrum was found to obey the berry robnik brody statistics there are no effects of dynamical tunneling in this regime due to the high energies as they decay exponentially with the inverse effective planck constant which is proportional to the square root of the energy

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

on spectral statistics of classically integrable systems this work is an extensive study of the spectral statistics of three representative classically integrable systems namely rectangle torus and circle billiards we analyse the statistics and focus on the related level spacing distribution p s and the delta statistics the agreement with the poissonian model is typically found to be perfect up to the outer unfolded energy scale beyond which the saturation is observed in agreement with berry s dynamical theory of the spectral rigidity where as the untypical systems are those where for example p s is not a smooth distribution but a sum of the delta functions due to the granularity of the energy scale for example in the rectangle with the rational squared sides ratio however even there we find reasonable trend towards poissonian statistics for large ranges l but we describe theoretically and numerically the broadening of the delta spikes when the rectangular billiard is slightly distorted away from a rational to an irrational shape and find excellent agreement also in irrational rectangle billiards we show and explain the existence of large fluctuations by one order of magnitude bigger than the statistical ones whose origin is in the closeness to some rational billiard shape these fluctuations and their amplitude are independent of the energy if the bin size shrinks inversely with energy finally we tested the mode distribution i e distribution of the reduced mode fluctuation number w and found that it was not poissonian in agreement with steiner s conjecture and in fact follows the prediction by bleher et al that its tail behaves as the general reason for non universal behaviour of p w is that at largest energy scales we are always in the saturation regime

wigner surmise for high order level spacing distributions of chaotic systems we suggest an extension of the wigner surmise for the nearest neighbor spacing distribution of energy levels of chaotic systems to include the nth order spacing distributions the main assumption is that the conditional probability density of occurrence of a level at a given distance from a fixed level provided that this distance contains n levels is expressed in terms of the n 1 th power of corresponding probability for a distance containing no levels at large spacings the nth order level distributions are assumed to have a gaussian shape as in the cases covered by the wigner surmise the expressions obtained are in good agreement with the results of numerical calculation by means of the random matrix theory for the three universal classes of symmetry

tunnelling between tori in phase space abstract this paper calculates the energy splitting e due to tunnelling between a pair of quantum states which correspond to classical motion on tori in phase space the calculation assumes that the states are separated in coordinate space however the result of the calculation e a 3 2 e 2 h has canonically invariant expressions for the tunnelling actions s and prefactor a and it is conjectured that the results apply even when the tori overlap in coordinate space this makes the results relevant to a very general problem connected with the einstein brillouin keller quantization scheme there is a surprising consequence of this calculation when the system is separable it is clear that e vanishes for most pairs of states this does not occur because the prefactor a vanishes but because the tunnelling action s is infinite as a parameter is varied to make the system separable s diverges logarithmically the results of this paper only apply to quasi integrable systems where s is finite p in the case of any exactly integrable system s is infinite and e vanishes in a more singular manner than e const h as 0

quantum localization of chaotic eigenstates and the level spacing distribution the phenomenon of quantum localization in classically chaotic eigenstates is one of the main issues in quantum chaos or wave chaos and thus plays an important role in general quantum mechanics or even in general wave mechanics in this work we propose two different localization measures characterizing the degree of quantum localization and study their relation to another fundamental aspect of quantum chaos namely the energy spectral statistics our approach and method is quite general and we apply it to billiard systems one of the signatures of the localization of chaotic eigenstates is a fractional power law repulsion between the nearest energy levels in the sense that the probability density to find successive levels on a distance s goes like proportionality s for small s where 0 1 and 1 corresponds to completely extended states we show that there is a clear functional relation between the exponent and the two different localization measures one is based on the information entropy and the other one on the correlation properties of the husimi functions we show that the two definitions are surprisingly linearly equivalent the approach is applied in the case of a mixed type billiard system m robnik j phys a math gen 16 3971 1983 in which the separation of regular and chaotic eigenstates is performed

beyond the berry robnik regime a random matrix study of tunneling effects we consider the energy spectra of quantum hamilton systems whose classical dynamics is of mixed type i e regular for some initial conditions in the classical phase space and chaotic for the complementary initial conditions in the strict semiclassical limit when the effective planck constant eff is sufficiently small the berry robnik br statistics applies while at larger values of eff or smaller energies one sees deviations from br due to localization and tunneling effects we derive a two level random matrix model which describes these effects and can be treated analytically in a closed form the coupling between the regular and chaotic levels due to tunneling is assumed to be gaussian distributed this two level model describes most of the features of matrices of large dimensions here n 1000 which we treat numerically and is predicted to apply in mixed type systems at low energies the proposed analytical level spacing distribution function has two parameters the br parameter characterizing the classical phase space and the coupling antenna distortion or tunneling parameter between states localization effects so far are not included in our analysis except in the special case where can be replaced by some effective eff

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

the level splitting distribution in chaos assisted tunnelling a compound tunneling mechanism from one integrable region to another mediated by a delocalized state in an intermediate chaotic region of phase space was recently introduced to explain peculiar features of tunneling in certain two dimensional systems this mechanism is known as chaos assisted tunneling we study its consequences for the distribution of the level splittings and obtain a general analytical form for this distribution under the assumption that chaos assisted tunneling is the only operative mechanism we have checked that the analytical form we obtain agrees with splitting distributions calculated numerically for a model system in which chaos assisted tunneling is known to be the dominant mechanism the distribution depends on two parameters the first gives the scale of the splittings and is related to the magnitude of the classically forbidden processes the second gives a measure of the efficiency of possible barriers to classical transportwhich may exist in the chaotic region if these are weak this latter parameter is irrelevant otherwise it sets an energy scale at which the splitting distribution crosses over from one type of behavior to another the detailed form of the crossover is also obtained and found to be in good agreement with numerical results for models for chaos assisted tunneling

universality in the flooding of regular islands by chaotic states we investigate the structure of eigenstates in systems with a mixed phase space in terms of their projection onto individual regular tori depending on dynamical tunneling rates and the heisenberg time regular states disappear and chaotic states flood the regular tori for a quantitative understanding we introduce a random matrix model the resulting statistical properties of eigenstates as a function of an effective coupling strength are in very good agreement with numerical results for a kicked system we discuss the implications of these results for the applicability of the semiclassical eigenfunction hypothesis

intermediate statistics in the regime of mixed classical dynamics we explain the arguments in support of the berry robnik 1984 17 2413 picture of the energy level statistics in the asymptotic strict far semiclassical limit of sufficiently small where the entire energy spectrum can be represented as a statistically independent superposition of regular and irregular level sequences the regular ones obeying the poissonian statistics and the irregular ones the rmt statistics goe or gue we generalize the results to describe not only the level spacing distribution the number variance and the delta statistics but also arbitrary statistics the probability to find k levels inside an interval of length l after unfolding very useful and effective approximations for are described we demonstrate very clearly that this regime is excellently described by this picture for k as high as the outer energy scale even when taking into account the regular component and only one the dominant chaotic component this we show numerically for the compactified standard map and for the quartic 2d billiard

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

evidence for the validity of the berry robnik surmise in a periodically pulsed spin system we study the statistical properties of the spectrum of a quantum dynamical system whose classical counterpart has mixed phase space structure consisting of two regular regions separated by a chaotic one we make use of a simple symmetry of the system to separate the eigenstates of the time evolution operator into two classes in agreement with the percival classification scheme we then use a method firstly developed by bohigas et al 1990 to evaluate the fractional measure of states belonging to the regular class and finally present the level spacing statistics for each class the level spacing distribution of states belonging to the irregular part of the spectra as well as that of the complete set of levels corroborate the berry robnik surmise we further present a statistical study of the regular levels the presence of intermediate states states which belong to neither class as long as r is finite phase spatially mixed among the set of regular ones together with the small fractional measure of regular states strongly affects the corresponding level spacing statistics resulting in a non negligible deviation from the expected poisson distribution we see however the remarkable agreement of the irregular level spacing statistics as a direct confirmation of the berry robnik surmise