shannon entropy at avoided crossings in the quantum transition from order to chaos shannon entropy is studied for the series of avoided crossings that characterize the transition from order to chaos in quantum mechanics in order to be able to study jointly this entropy for discrete and continuous probability calculations have been performed on a quantized map the kicked harper map resulting in a different behavior as order chaos transition takes place for the discrete position representation and continuous coherent state representation cases this different behavior is analyzed in terms of the distribution of zeros of the husimi function

free vibration analysis of stochastic strings the free vibration of strings with randomly varying mass and stiffness is considered the joint probability density functions of the eigenvalues and eigenfunctions are characterized in terms of the solution of a pair of stochastic non linear initial value problems analytical solutions of these equations based on the method of stochastic averaging are obtained the effects of the mean and autocorrelation of the mass process are included in the analysis numerical results for the marginal probability density functions of eigenvalues and eigenfunctions are obtained and are found to compare well with monte carlo simulation results the random eigenvalues when normalized with respect to their corresponding deterministic values are observed to tend to become first order stochastically stationary with respect to the mode count

correlations between zeros of a random polynomial we obtain exact analytical expressions for correlations between real zeros of the kac random polynomial we show that the zeros in the interval 1 1 are asymptotically independent of the zeros outside of this interval and that the straightened zeros have the same limit translation invariant correlations then we calculate the correlations between the straightened zeros of theo 1 random polynomial

quantum variance and ergodicity for the baker s map we prove an egorov theorem or quantum classical correspondence for the quantised baker s map valid up to the ehrenfest time this yields a logarithmic upper bound for the decay of the quantum variance and as a corollary a quantum ergodic theorem for this map

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

equipartition of the eigenfunctions of quantized ergodic maps on the torus we give a simple proof of the equipartition of the eigenfunctions of a class of quantized ergodic area preserving maps on the torus examples are the irrational translations the skew translations the hyperbolic automorphisms and some of their perturbations

chaos revealing multiplicative representation of quantum eigenstates the quantisation of the two dimensional toric and spherical phase spaces is considered in analytic coherent state representations every pure quantum state admits therein a finite multiplicative parametrisation by the zeros of its husimi function for eigenstates of quantised systems this description explicitly reflects the nature of the underlying classical dynamics in the semiclassical regime the distribution of the zeros in the phase space becomes one dimensional for integrable systems and highly spread out conceivably uniform for chaotic systems this multiplicative representation thereby acquires a special relevance for semiclassical analysis in chaotic systems

on the nullity of the line graph of unicyclic graph with depth one abstract a connected graph with a unique cycle is called a unicyclic graph a unicyclic graph with depth one may be thought of as being obtained from a cycle by appending n i pendent edges on each vertex v i in the cycle c t for some integer t 3 denoted by c n 1 n 2 n t in this paper we give a complete characterization on the nullity of the line graph g l c n 1 n 2 n t as follows let m i n i 0 then i g 2 if and only if m 0 and t 0 mod 4 ii g 1 if and only if m 1 and either 1 n i 0 1 for i 1 t the length of any zero chain of n 1 n 2 n t is even and t m 0 mod 4 or 2 t 0 mod 4 and one of n 1 n 3 n t 1 0 and n 2 n 4 n t 0 must hold iii g 0 otherwise

the berry phase for spin in the majorana representation a continuous cyclic sequence of quantum states has an associated geometric or berry phase for spin j such a sequence is described by a cyclic change in the 2j 1 coefficients of the basis states the berry phase is analysed here for the general case that is the coefficients are allowed to vary in an arbitrary cyclic manner the result is expressed in geometric terms specifically in the democratic representation due to majorana this uniquely characterizes the spin state up to overall phase by the positions of 2j dots on the unit sphere of directions in real space if the positions are denoted by unit vectors where each traces out a parametrized loop on the sphere and the berry phase is given by an integral of combinations of these vectors

reliability analysis of randomly vibrating structures with parameter uncertainties the problem of reliability analysis of randomly driven stochastically parametered vibrating structures is considered the excitation components are assumed to be jointly stationary gaussian random processes the randomness in structural parameters is modelled through a vector of mutually correlated non gaussian random variables the structure is defined to be safe if a set of response quantities remain within prescribed thresholds over a given duration of time an approximation for the failure probability conditioned on structure randomness is first formulated in terms of recently developed multivariate extreme value distributions an estimate of the unconditional failure probability is subsequently obtained using a method based on taylor series expansion this has involved the evaluation of gradients of the multivariate extreme value distribution conditioned on structure random variables with respect to the basic variables analytical and numerical techniques for computing these gradients have been discussed issues related to the accuracy of the proposed method have been examined the proposed method has applications in reliability analysis of vibrating structural series systems where different failure modes are mutually dependent illustrative examples presented include the seismic fragility analysis of a piping network in a nuclear power plant

on the zeros of the husimi distribution the basic features of the zeros of the husimi phase space density of a quantum eigenstate are discussed it is demonstrated that some special properties of the harmonic oscillator related to the number of zeros and their location within the classical energy shell are not valid for anharmonic potentials as illustrated by numerical examples for the morse oscillator also discussed is the distribution of zeros in quantum poincare sections for systems with two degrees of freedom

on the nullity of a graph with cut points let g be a simple graph of order n and a g be its adjacency matrix the nullity of a graph g denoted by g is the multiplicity of the eigenvalue zero in the spectrum of a g denote by ck and lk the set of all connected graphs with k induced cycles and the set of line graphs of all graphs in ck respectively in 1998 sciriha i sciriha on singular line graphs of trees congr numer 135 1998 73 91 show that the order of every tree whose line graph is singular is even then gutman and sciriha i gutman i sciriha on the nullity of line graphs of trees discrete math 232 2001 35 45 show that the nullity set of l0 is 0 1 in this paper we investigate the nullity of graphs with cut points and deduce some concise formulas then we generalize scirihas result showing that the order of every graph g is even if such a graph g satisfies that g ck and l g k 1 and the nullity set of lk is 0 1 k k 1 for any given k where l g denotes the line graph of the graph g

the nullity of k 1 cyclic graphs abstract a simple connected graph is called a k cyclic graph provided that the number of edges equals its order plus k 1 in this paper we define a class of k 1 cyclic graph denoted by k the singularity of graph is studied also the k 1 cyclic graphs that contain graph as an induced subgraph are paid much attention we characterize those graphs that have extreme nullities the results generalize the corresponding theorems on unicyclic and bicyclic graphs

a characterization of graphs with rank 5 abstract the rank of a graph g is defined to be the rank of its adjacency matrix in this paper we consider the following problem what is the structure of a connected graph g with rank 5 or equivalently what is the structure of a connected n vertex graph g whose adjacency matrix has nullity n 5 in this paper we completely characterize connected graphs g whose adjacency matrix has rank 5

on the nullity of bipartite graphs abstract the nullity of a graph is defined to be the multiplicity of the eigenvalue zero in the spectrum of the adjacency matrix of the graph in this paper we obtain the nullity set of bipartite graphs of order n and characterize the bipartite graphs with nullity n 4 and the regular bipartite graphs with nullity n 6

trees with maximum nullity abstract the nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum among n vertex trees the star has greatest nullity equal to n 2 we generalize this by showing that among n vertex trees whose vertex degrees do not exceed a certain value d the greatest nullity is n 2 n 1 d methods for constructing such maximum nullity trees are described

weak coherent states related to the multiplicative group mathbb c generalized coherent states which are associated with the multiplicative group of non zero complex numbers are introduced they have the property of temporal stability and they form a total set in some subspace of the full hilbert space they are highly non classical states in the sense that in general their bargmann functions have zeros which are related to regions where their wigner functions take negative values states with bargmann functions sin az and exp bz az k are examples of the general formalism and they are discussed in detail

random analytic chaotic eigenstates the statistical properties of random analytic functions z are investigated as a phase space model for eigenfunctions of fully chaotic systems we generalize to the plane and to the hyperbolic plane a theorem concerning the equidistribution of the zeros of z previously demonstrated for a spherical phase space su 2 polynomials for systems with time reversal symmetry the number of real roots is computed for the three geometries in the semiclassical regime the local correlation functions are shown to be universal independent of the system considered or the geometry of phase space in particular the autocorrelation function of is given by a gaussian function the connections between this model and the gaussian random function hypothesis as well as the random matrix theory are discussed

phase space approach to quantum dynamics replaces the schrodinger equation for the time propagation of states of a quantized 2d spherical phase space by the dynamics of a system of n particles lying in phase space this is done through factorization formulae of an analytic function theory arising in a coherent state representation the particles being the zeroes of the quantum state for linear hamiltonians like a spin in a uniform magnetic field the motion of the particles is classical however nonlinear terms induce interactions between the particles their time propagation is studied and it is shown that contrary to integrable systems for chaotic maps they tend to fill as their classical counterpart the whole phase space in a uniform way

chaotic analytic zero points exact statistics for those of a random spin state a natural statistical ensemble of 2j points on the unit sphere can be associated via the majorana representation with a random quantum state of spin j and an exact expression is obtained here for the general k point correlation function in this ensemble the pair correlation in the large j limit takes the relatively simple form where and is the angular separation of the pair of points on the sphere it appears from the numerical work of others that in this limit these statistics are typical of the zero points of analytic functions associated with chaotic quantum dynamical systems

topological properties of quantum periodic hamiltonians we consider periodic quantum hamiltonians on the torus phase space harper like hamiltonians we calculate the topological chern index which characterizes each spectral band in the generic case this calculation is made by a semiclassical approach with the use of quasi modes as a result the chern index is equal to the homotopy of the path of these quasi modes on phase space as the floquet parameter of the band is varied it is quite interesting that the chern indices defined as topological quantum numbers can be expressed from simple properties of the classical trajectories

renyi wehrl entropies as measures of localization in phase space we generalize the concept of the wehrl entropy of quantum states which gives a basis independent measure of their localization in phase space we discuss the minimal values and the typical values of these r enyi wehrl entropies for pure states for spin systems according to lieb s conjecture the minimal values are provided by the spin coherent states though lieb s conjecture remains unproven we give new proofs of partial results that may be generalized for other systems we also investigate random pure states and calculate the mean r enyi wehrl entropies averaged over the natural measure in the space of pure quantum states

positive and negative inertia index of a graph abstract in this paper the positive and negative inertia index of trees unicyclic graphs and bicyclic graphs are discussed the methods of calculating them are obtained and an inequality about the difference between positive and negative inertia index is proved moreover a conjecture is proposed

nonsingular vertex weighted block graphs abstract a graph g is nonsingular singular if its adjacency matrix a g is nonsingular singular in this article we consider the nonsingularity of block graphs i e graphs in which every block is a clique extending the problem we characterize nonsingular vertex weighted block graphs in terms of reduced vertex weighted graphs resulting after successive deletion and contraction of pendant blocks special cases where nonsingularity of block graphs may be directly determined are discussed

fractional monodromy in systems with coupled angular momenta we present a one parameter family of systems with fractional monodromy which arises from a 1 2 diagonal action of a dynamical symmetry so 2 i n a regime of adiabatic separation of slow and fast motions we relate the presence of fractional monodromy to a redistribution of states both in the quantum and in the semi quantum spectra

chaotic eigenfunctions in phase space we study individual eigenstates of quantized area preserving maps on the 2 torus which are classically chaotic in order to analyze their semiclassical behavior we use the bargmann husimi representations for quantum states as well as their stellar parametrization which encodes states through a minimal set of points in phase space the constellation of zeros of the husimi density we rigorously prove that a semiclassical uniform distribution of husimi densities on the torus entails a similar equidistribution for the corresponding constellations we deduce from this property a universal behavior for the phase patterns of chaotic bargmann eigenfunctions which is reminiscent of the wkb approximation for eigenstates of integrable systems though in a weaker sense in order to obtain more precise information on chaotic eigenconstellations we then model their properties by ensembles of random states generalizing former results on the 2 sphere to the torus geometry this approach yields statistical predictions for the constellations which fit quite well the chaotic data we finally observe that specific dynamical information e g the presence of high peaks like scars in husimi densities can be recovered from the knowledge of a few long wavelength fourier coefficients which therefore appear as valuable order parameters at the level of individual chaotic eigenfunctions

a characterization of graphs with rank 4 the rank of a graph g is defined to be the rank of its adjacency matrix in this paper we consider the following problem what is the structure of a connected graph with rank 4 this question has not yet been fully answered in the literature and only some partial results are known in this paper we resolve this question by completely characterizing graphs g whose adjacency matrix has rank 4

quantum chaotic dynamics and random polynomials we investigate the distribution of roots of polynomials of high degree with random coefficients which among others appear naturally in the context of quantum chaotic dynamics it is shown that under quite general conditions their roots tend to concentrate near the unit circle in the complex plane in order to further increase this tendency we study in detail the particular case of self inversive random polynomials and show that for them a finite portion of all roots lies exactly on the unit circle correlation functions of these roots are also computed analytically and compared to the correlations of eigenvalues of random matrices the problem of ergodicity of chaotic wavefunctions is also considered for that purpose we introduce a family of random polynomials whose roots spread uniformly over phase space while these results are consistent with random matrix theory predictions they provide a new and different insight into the problem of quantum ergodicity special attention is devoted to the role of symmetries in the distribution of roots of random polynomials

on the nullity of graphs with pendant trees the nullity of a graph is defined as the multiplicity of the eigenvalue zero in the spectrum of the adjacency matrix of the graph we investigate a class of graphs with pendant trees and express the nullity of such graph in terms of that of its subgraphs as an application of our results we characterize unicyclic graphs with a given nullity

the chaotic analytic function in contrast to a stationary gaussian random function of a real variable which is free to have any correlation function the closest analogous analytic random function in the complex plane has no true freedom it is statistically unique since it has arisen only recently as an apparently universal feature in the physical context of quantum chaos i refer to it here as the chaotic analytic function i note that it is implied by the assumption that a quantum chaotic wavefunction has gaussian randomness and has a constant value for the average of its wigner function in phase space interpreted literally this shows that the chaotic analytic function is the bargmann function of a pure white noise wavefunction more physically if constant is replaced by smooth on the scale of a planck area these assumptions are the semiclassical ones made by berry for chaotic eigenstates the analysis shows that the chaotic analytic function is still obtained semiclassically

eigenfunctions of non integrable systems in generalised phase spaces the authors use the generalised coherent state distribution to study in phase space the eigenfunctions of non integrable hamiltonians the methods available in the weyl group the husimi distribution are extended to the more complicated phase spaces that arise from other lie groups they demonstrate the feasibility of numerical calculations in two models with w2 and su 3 symmetries emphasising the similarities and showing the emergence of classical invariant structures in the semiclassical limit

on the nullity of bicyclic graphs the nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum in this paper we obtain the nullity set of n vertex unicyclic graphs and characterize the unicyclic graphs with extremal nullity

wigner function statistics in classically chaotic systems we have studied statistical properties of the values of the wigner function w x of 1d quantum maps on compact 2d phase space of finite area v for this purpose we have defined a wigner function probability distribution p w 1 v w w x dx which has by definition fixed first and second moments in particular we concentrate on relaxation of time evolving quantum states in terms of w x starting from a coherent state we have shown that for a classically chaotic quantum counterpart the distribution p w in the semiclassical limit becomes a gaussian distribution that is fully determined by the first two moments numerical simulations have been performed for the quantum sawtooth map and the quantized kicked top in a quantum system with hilbert space dimension n 1 the transition of p w to a gaussian distribution was observed at times t log n in addition it has been shown that the statistics of wigner functions of propagator eigenstates is gaussian as well in the classically fully chaotic regime we have also studied the structure of the nodal cells of the wigner function in particular the distribution of intersection points between the zero manifold and arbitrary straight lines

classical limit of the quantized hyperbolic toral automorphisms the canonical quantization of any hyperbolic symplectomorphisma of the 2 torus yields a periodic unitary operator on an dimenional hilbert space n 1 h we prove that this quantum system becomes ergodic and mixing at the classical limit n n prime which can be interchanged with the time average limit the recovery of the stochastic behaviour out of a periodic one is based on the same mechanism under which the uniform distribution of the classical periodic orbits reproduces the lebesgue measure the wigner functions of the eigenstates supported on the classical periodic orbits are indeed proved to become uniformly speread in phase space

vortex structure and characterization of quasiperiodic functions quasiperiodic functions qpfs are characterized by their full vortex structure in one unit cell this characterization is much finer and more sensitive than the topological one given by the total vorticity per unit cell the chern index it is shown that qpfs with an arbitrarily prescribed vortex structure exist by constructing explicitly such a standard qpf two qpfs with the same vortex structure are equivalent in the sense that their ratio is a function which is strictly periodic nonvanishing and at least continuous a general qpf can then be approximately reconstructed from its vortex structure on the basis of the standard qpf and the equivalence concept as another application of this concept a simple method is proposed for calculating the quasiperiodic eigenvectors of periodic matrices possible applications to the quantum chaos problem on a phase space torus are briefly discussed

on the nullity of graphs with pendent vertices abstract the nullity of a graph g denoted by g is the multiplicity of the eigenvalue zero in its spectrum cheng and liu b cheng b liu on the nullity of graphs electron j linear algebra 16 2007 60 67 characterized the extremal graphs attaining the upper bound n 2 and the second upper bound n 3 in this paper as the continuance of it we determine the extremal graphs with pendent vertices achieving the third upper bound n 4 and fourth upper bound n 5 we then proceed recursively to construct all graphs with pendent vertices which satisfy g 0 our results provide a unified approach to determine n vertex unicyclic respectively bicyclic and tricyclic graphs which achieve the maximal and second maximal nullity and characterize n vertex extremal trees attaining the second and third maximal nullity as a consequence we respectively determine the nullity sets of trees unicyclic graphs bicyclic graphs and tricyclic graphs on n vertices

a characterization of long graphs of arbitrary rank abstract the rank of a graph g is defined to be the rank of its adjacency matrix a connected graph g of rank n is called long if it has maximum diameter among all connected graphs with rank n in this paper we characterize the long graphs of rank n for n an arbitrary positive integer

on the distribution of zeros of chaotic wavefunctions wavefunctions in a phase space representation can be expressed as entire functions of their zeros if the phase space is compact these zeros seem to hide a lot of relevant and explicit information about the underlying classical dynamics besides an understanding of their statistical properties may prove to be useful in the analytical calculations of the wavefunctions in quantum chaotic systems this motivates us to pursue the present study which by a numerical statistical analysis shows that both long range correlations as well as short range correlations exist between zeros while the latter turn out to be universal and parametric independent the former seem to be system dependent and are significantly affected by various parameters i e symmetry localization etc furthermore for the delocalized quantum dynamics the distribution of these zeros seem to mimic that of the zeros of the random functions as well as random polynomials

periodic orbit basis for the quantum baker map a set of quantum states dynamically related to the classical periodic orbits of a chaotic map is used as a basis in which the description of the eigenstates of its quantum version is greatly simplified this set can be improved with the inclusion of short time propagation along the stable and unstable manifolds of the periodic orbits resulting in a construction similar to the scar functions of vergini j phys a 33 4709 2000 the average participation ratio is used to quantify the quality of the basis

the inertia of unicyclic graphs and the implications for closed shells the inertia of a graph is an integer triple specifying the number of negative zero and positive eigenvalues of the adjacency matrix of the graph a unicyclic graph is a simple connected graph with an equal number of vertices and edges this paper characterizes the inertia of a unicyclic graph in terms of maximum matchings and gives a linear time algorithm for computing it chemists are interested in whether the molecular graph of an unsaturated hydrocarbon is properly closed shell having exactly half of its eigenvalues greater than zero because this designates a stable electron configuration the inertia determines whether a graph is closed shell and hence the reported result gives a linear time algorithm for determining this for unicyclic graphs

analytic representations in quantum mechanics various euclidean hyperbolic and elliptic analytic representations are introduced and relations among them are discussed the bargmann analytic representation in the complex plane is considered and its relation to other phase space methods for the harmonic oscillator is reviewed the general theory that relates the growth of analytic functions with the density of their zeros is applied to bargmann functions and it leads to theorems on the completeness of sequences of glauber coherent states two hyperbolic analytic representations in the unit disc based on su 1 1 coherent states and also on phase states are introduced a third analytic representation in the complex plane based on barut girardello states is also considered and transformations which relate it to the other ones are studied in the case of systems with finite dimensional hilbert space an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced the berezin formalism in the euclidean hyperbolic and elliptic cases is discussed contour analytic representations in these three cases are also presented

exact statistics of systems with uncertainties an analytical theory of rank one stochastic dynamic systems this paper concerns the response of uncertain vibro acoustic and structural dynamic systems here exact expressions are presented for the statistics of systems with a random rank one component the expressions are derived using the sherman morrison update formula that gives the exact expression of the disturbed response for any magnitude of the disturbance it is shown that the probability density function pdf of any transfer function is a simple function of the pdf of the disturbance magnitude of the random component the expressions for the mean variance and covariance of any transfer function and at any frequency of a random system necessitate non trivial integrals exact including closed form expressions of these integrals are derived in the particular cases of a real or complex normal disturbance magnitude and qualitative differences between these two cases are highlighted the theoretical and practical advantages of the theory are discussed and applied to a model of a bladed disk subjected to random damage the comparison with monte carlo simulations demonstrates that the statistics can be evaluated efficiently and precisely the theory derived formally in the context of discretised systems is directly applicable to continuous systems

on the eigenvalues of distance powers of circuits abstract taking the d th distance power of a graph one adds edges between all pairs of vertices of that graph whose distance is at most d it is shown that only the numbers 3 2 1 0 1 2 d can be integer eigenvalues of a circuit distance power moreover their respective multiplicities are determined and explicit constructions for corresponding eigenspace bases containing only vectors with entries 1 0 1 are given

on the null spaces of acyclic and unicyclic singular graphs for acyclic and unicyclic graphs we determine a necessary and sufficient condition for a graph g to be singular further it is shown that this characterization can be used to construct a basis for the null space of g

the monge metric on the sphere and geometry of quantum states topological and geometrical properties of the set of mixed quantum states in the n dimensional hilbert space are analysed assuming that the corresponding classical dynamics takes place on the sphere we use the vector su 2 coherent states and the generalized husimi distributions to define the monge distance between two arbitrary density matrices the monge metric has a simple semiclassical interpretation and induces a non trivial geometry among all pure states the distance from the maximally mixed state proportional to the identity matrix admits the largest value for the coherent states while the delocalized chaotic states are close to this contrasts the geometry induced by the standard trace hilbert schmidt or bures metrics for which the distance from is the same for all pure states we discuss possible physical consequences including unitary time evolution and the process of decoherence we introduce also a simplified monge metric defined in the space of pure quantum states and more suitable for numerical computation

phase retrieval problem in quantum chaos and its relation to the origin of irreversibility i abstract the purpose of this paper are i to study characteristics features of phase in quantum dynamics corresponding to chaos and ii to discuss implication of these features for the origin of irreversibility the first is done by focusing attention on topological defects of phase in the husimi representation we find that the process of generating topological defects corresponds to the folding of morse smale horseshoe dynamics moreover the distribution of topological defects changes from regular to random under the time evolution this change is a distinctive feature of quantum dynamics corresponding to chaos the second is done in terms of phase retrieval problem we present numerical results which strongly suggest the following the randomness in the distribution of topological defects is the cause of difficulty in retrieving phase based on these results we present the following hypothesis the loss of phase information in statistical mechanics can be understood in terms of the randomness of phase in quantum dynamics corresponding to chaos

phase space flow in the husimi representation we derive a continuity equation for the husimi function evolving under a general non hermitian hamiltonian and identify the phase space flow associated with it for the case of unitary evolution we obtain explicit formulas for the quantum flow which can be written as a classical term plus quantum corrections the quantum terms can be expanded in powers of planck s constant providing a series of semiclassical corrections to the classical flow we test the exact and semiclassical formulas for a particle in a double well potential and find numerical evidence that the zeros of the husimi function are always saddle points of the flow merging or splitting of stagnation points reported for the wigner flow was not observed

exact statistical properties of the zeros of complex random polynomials the zeros of complex gaussian random polynomials with coefficients such that the density in the underlying complex space is uniform are known to have the same statistical properties as the zeros of the coherent state representation of one dimensional chaotic quantum systems we extend the interpretation of these polynomials by showing that they also arise as the wavefunction for a quantum particle in a magnetic field constructed from a random superposition of states in the lowest landau level a study of the statistical properties of the zeros is undertaken using exact formulae for the one and two point distribution functions attention is focused on the moments of the two point correlation in the bulk the variance of a linear statistic and the asymptotic form of the two point correlation at the boundary a comparison is made with the same quantities for the eigenvalues of complex gaussian random matrices

intrinsic ordering of quasienergy states for mixed regular chaotic quantum systems zeros of the husimi distribution a method is proposed which allows an intrinsic ordering of the quasienergy states for a time periodic quantum system according to the number of essential zeros of the husimi distribution in phase space the corresponding classical system shows a typical mixed dynamics with coexisting regular and chaotic regions in phase space the ordering of states allows a straightforward comparison with semiclassical quantities moreover a direct connection of the organization of zeros with the underlying classical phase space dynamics is demonstrated

time reversal symmetry and random polynomials we analyse the density of roots of random polynomials where each complex coefficient is constructed of a random modulus and a fixed deterministic phase the density of roots is shown to possess a singular component only in the case for which the phases increase linearly with the index of coefficients this means that contrary to earlier belief eigenvectors of a typical quantum chaotic system with some antiunitary symmetry will not display a clustering curve in the stellar representation moreover a class of time reverse invariant quantum systems is shown for which spectra display fluctuations characteristic of orthogonal ensemble while eigenvectors confer to predictions of unitary ensemble