production rate of the coarse grained gibbs entropy and the kolmogorov sinai entropy a real connection we discuss the connection between the kolmogorov sinai entropy hks and the production rate of the coarse grained gibbs entropy rg detailed numerical computations show that the often accepted identification of the two quantities does not hold in systems with intermittent behavior and or very different characteristic times and in systems presenting pseudochaos the basic reason for this is in the asymptotic with respect to time nature of hks while rg is a quantity related to short time features of a system

fractional moment methods for anderson localization with saw representation the green function contains much information about physical systems mathematically the fractional moment method fmm developed by aizenman and molchanov connects the green function and the transport of electrons in the anderson model recently it has been discovered that the green function on a graph can be represented using self avoiding walks on a graph which allows us to connect localization properties in the system and graph properties we discuss fmm in terms of the self avoiding walks on a general graph the only general condition being that the graph has a uniform bound on the vertex degree

functionals of brownian motion localization and metric graphs we review several results related to the problem of a quantum particle in a random environment in an introductory part we recall how several functionals of brownian motion arise in the study of electronic transport in weakly disordered metals weak localization two aspects of the physics of the one dimensional strong localization are reviewed some properties of the scattering by a random potential time delay distribution and a study of the spectrum of a random potential on a bounded domain the extreme value statistics of the eigenvalues then we mention several results concerning the diffusion on graphs and more generally the spectral properties of the schrodinger operator on graphs the interest of spectral determinants as generating functions characterizing the diffusion on graphs is illustrated finally we consider a two dimensional model of a charged particle coupled to the random magnetic field due to magnetic vortices we recall the connection between spectral properties of this model and winding functionals of planar brownian motion

asymptotic analysis of the lyapunov exponent and rotation number of the random oscillator and application we construct asymptotic expansions for the exponential growth rate lyapunov exponent and rotation number of the random oscillator when the noise is large small rapidly varying or slowly varying we then apply our results to problems in the stability of the random oscillator the spectrum of the one dimensional random schr6dinger operator and wave propagation in a one dimensional random medium

spectral flow and level spacing of edge states for quantum hall hamiltonians we consider a non relativistic particle on the surface of a semi infinite cylinder of circumference l submitted to a perpendicular magnetic field of strength b and to the potential of impurities of maximal amplitude w this model is of importance in the context of the integer quantum hall effect in the regime of strong magnetic field or weak disorder b w it is known that there are chiral edge states which are localized within a few magnetic lengths close to and extended along the boundary of the cylinder and whose energy levels lie in the gaps of the bulk system these energy levels have a spectral flow uniform in l as a function of a magnetic flux which threads the cylinder along its axis through a detailed study of this spectral flow we prove that the spacing between two consecutive levels of edge states is bounded below by 2 l 1 with 0 independent of l and of the configuration of impurities this implies that the level repulsion of the chiral edge states is much stronger than that of extended states in the usual anderson model and their statistics cannot obey one of the gaussian ensembles our analysis uses the notion of relative index between two projections and indicates that the level repulsion is connected to topological aspects of quantum hall systems

a numerical study of sparse random matrices a numerical study is presented for the eigensolution statistics of largen n real and symmetric sparse random matrices as a function of the mean numberp of nonzero elements per row the model shows classical percolation and quantum localization transitions atp c 1 andp q 1 respectively in the rigid limitp n we demonstrate that the averaged density of states follows the wigner semicircle law and the corresponding nearest energy level spacing distribution functionp s obeys the wigner surmise in the very sparse matrix limitp n withp p q a singularity e 1 e is found as e 0 and exponential tails develop in the high e regions but thep s distribution remains consistent with level repulsion the localization properties of the model are examined by studying both the eigenvector amplitude and the density fluctuations the valuep q 1 4 is roughly estimated in agreement with previous studies of the anderson transition in dilute bethe lattices

localization and spectral statistics in a banded random matrix ensemble we investigate the localization properties of the eigenvectors of a banded random matrix ensemble in which the diagonal matrix elements increase along the diagonal we relate the results to a transition in the spectral statistics which is observed as a parameter is vaned and discuss the relevance of this model to the quantum mechanics of chaotic hamiltonian systems

suboptimal minimum cluster volume cover based method for measuring fractal dimension a new method for calculating fractal dimension is developed in this paper the method is based on the box dimension concept however it involves direct estimation of a suboptimal covering of the data set of interest by finding a suboptimal cover this method is better able to estimate the required number of covering elements for a given cover size than is the standard box counting algorithm moreover any decrease in the error of the covering element count directly increases the accuracy of the fractal dimension estimation in general our method represents a mathematical dual to the standard box counting algorithm by not solving for the number of boxes used to cover a data set given the size of the box instead the method chooses the number of covering elements and then proceeds to find the placement of smallest hyperellipsoids that fully covers the data set this method involves a variant of the fuzzy c means clustering algorithm as well as the use of the minimum cluster volume clustering algorithm a variety of fractal dimension estimators using this suboptimal covering method are discussed finally these methods are compared to the standard box counting algorithm and wavelet decomposition methods for calculating fractal dimension by using one dimensional cantor dust sets and a set of standard brownian random fractal images

quantum diffusion in the generalized harper equation we study numerically the dynamic and spectral properties of a one dimensional quasi periodic system where site energies are given by with denoting the kth quasiperiodic lattice site when is given by the reciprocal lattice vector with n and m being successive fibonacci numbers the variance of the wavepacket is found to grow quadratically in time regardless of the potential strength v for other values of f there exists a critical value of v beyond which the growth of the wavepacket variance is bounded in particular an anomalous diffusion takes place for corresponding to with generic integers m and n the level spacing distribution is also examined and the corresponding exponent is observed to decrease with v

spectral analysis of selfadjoint jacobi matrices with periodically modulated entries this paper deals with the spectral analysis of a class of selfadjoint unbounded jacobi matrices j with modulated entries the entries have the form of smooth sequences that increase to infinity multiplied by proper periodic sequences for this class criteria for pure absolute continuity of the spectrum or its discreteness and the asymptotics of generalized eigenvectors of j are given some examples illustrating the stability zones of spectral structure are presented

statistics of quasienergies in chaotic and random systems abstract the statistics of quasienergies are analyzed for periodically driven chaotic systems and found to be similar to those of truly random models these differ from the results that were obtained so far for chaotic systems with time independent hamiltonians the separations of the quasienergies and the 3 statistic are calculated numerically for chaotic as well as for truly random models local statistical measures are introduced in order to investigate the repulsion of quasienergies the results provide further evidence for anderson localization in chaotic systems with hamiltonians that are periodic in time

rank one perturbations and the zeros of paraorthogonal polynomials on the unit circle we prove several results about zeros of paraorthogonal polynomials using the theory of rank one perturbations of unitary operators in particular we obtain new details on the interlacing of zeros for successive popuc

localization and universality of poisson statistics for the multidimensional anderson model at weak disorder we prove anderson localization with the mean field lyapunov exponent and poisson statistics for eigenvalue spacing for the multi dimensional anderson model at weak disorder these results are obtained by developing the supersymmetric formalism initiated in w1 see also sjw rid

generating functions and elementary young tableaux when constructing polynomial bases for group representations one is led naturally to consider a stretched product of states elementary multiplets are states that cannot be expressed as the stretched product of any other two states and they can be identified with terms occurring in certain generating functions in this paper the authors define an analogous stretched product of young tableaux together with elementary tableaux these may be used in a manner similar to elementary multiplets to construct generating functions they consider several examples where this method can be applied and in particular give new large n rank free branching rule generating functions for the group subgroup pairs su n su m contained in implied by su n m first four labels non zero so n contained in implied by su n first three labels non zero and sp 2n contained in implied by su 2n first five labels non zero this method of finding these generating functions uses the theory of grobner bases which allows the authors to make precise the notion of compatibility relations or forbidden products of elementary tableaux

on subordinacy and analysis of the spectrum of schrodinger operators with two singular endpoints the theory of subordinacy is extended to all one dimensional schrodinger operatorsfor which the corresponding differential expression l d 2 dr 2 v r is in the limit point case at both ends of an interval a b with v r locally integrable this enables a detailed classification of the absolutely continuous and singular spectra to be established in terms of the relative asymptotic behaviour of solutions of lu xu x e as r a and r b the result provides a rigorous but straightforward method of direct spectral analysis which has very general application and somefurther properties of the spectrum are deduced from the underlying theory

universality limits of a reproducing kernel for a half line schrodinger operator and clock behavior of eigenvalues we extend some recent results of lubinsky levin simon and totik from measures with compact support to spectral measures of schrodinger operators on the half line in particular we define a reproducing kernel sl for schrodinger operators and we use it to study the fine spacing of eigenvalues in a box of the half line schrodinger operator with perturbed periodic potential we show that if solutions u x are bounded in x by e epsilon x uniformly for near the spectrum in an average sense and the spectral measure is positive and absolutely continuous in a bounded interval i in the interior of the spectrum with xi 0 in i then uniformly in i frac s l xi 0 a l xi 0 b l s l xi 0 xi 0 rightarrow frac sin pi rho xi 0 a b pi rho xi 0 a b where d is the density of states we deduce that the eigenvalues near 0 in a large box of size l are spaced asymptotically as frac 1 l rho we adapt the methods used to show similar results for orthogonal polynomials

sum rules and spectral measures of schrodinger operators with l 2 potentials necessary and sufficient conditions are presented for a positive measure to be the spectral measure of a half line schrodinger operator with square integrable potential

the rotation number for almost periodic potentials we define and analyze the rotation number for the almost periodic schrodinger operatorl d2 dx2 q x we use the rotation number to discuss i the spectrum ofl ii its relation to the korteweg de vries equation

local fluctuation of the spectrum of a multidimensional anderson tight binding model we consider the anderson tight binding modelh v acting inl 2 z d and its restrictionh to finite hypercubes z d herev v x x z d is a random potential consisting of independent identically distributed random variables let e j j be the eigenvalues ofh and let j e e j e j 1 be its rescaled eigenvalues then assuming that the exponential decay of the fractional moment of the green function holds for complex energies neare and that the density of statesn e exists ate we shall prove that the random sequence j e j considered as a point process onr 1 converges weakly to the stationary poisson point process with intensity measuren e dx as gets large thus extending the result of molchanov proved for a one dimensional continuum random schrodinger operator on the other hand the exponential decay of the fractional moment of the green function was established recently by aizenman molchanov and graf as a technical lemma for proving anderson localization at large disorder or at extreme energy thus our result in this paper can be summarized as follows near the energye where anderson localization is expected there is no correlation between eigenvalues ofh if is large

dynamical upper bounds on wavepacket spreading we derive a general upper bound on the spreading rate of wavepackets in the framework of schrodinger time evolution our result consists of showing that a portion of the wavepacket cannot escape outside a ball whose size grows dynamically in time where the rate of this growth is determined by properties of the spectral measure and by spatial properties of solutions of an associated time independent schrodinger equation we also derive a new lower bound on the spreading rate which is strongly connected with our upper bound we apply these new bounds to the fibonacci hamiltonian the most studied one dimensional model of quasicrystals as a result we obtain for this model upper and lower dynamical bounds establishing wavepacket spreading rates which are intermediate between ballistic transport and localization the bounds have the same qualitative behavior in the limit of large coupling

the local structure of the spectrum of the one dimensional schrodinger operator let h v frac d text 2 dt text 2 q t omega be an one dimensional random schrodinger operator in 2 v v with the classical boundary conditions the random potentialq t has a formq t f x t wherex t is a brownian motion on the compact riemannian manifoldk andf k r 1 is a smooth morse function mathop min limits k f 0 let n v delta sum limits ei v in delta 1 where 0 e i v are the eigenvalues ofh v the main result theorem 1 of this paper is the following ifv e 0 0 k z anda 0 a is a fixed constant then p left n v left e 0 frac a 2v e 0 frac a 2v right k right xrightarrow v to infty e an e 0 an e 0 k k wheren e 0 is a limit state density ofh v v this theorem mean that there is no repulsion between energy levels of the operatorh v v the second result theorem 2 describes the phenomen of the repulsion of the corresponding wave functions

delocalization in random polymer models a random polymer model is a one dimensional jacobi matrix randomly composed of two finite building blocks if the two associated transfer matrices commute the corresponding energy is called critical such critical energies appear in physical models an example being the widely studied random dimer model it is proven that the lyapunov exponent vanishes quadratically at a generic critical energy and that the density of states is positive there large deviation estimates around these asymptotics allow to prove optimal lower bounds on quantum transport showing that it is almost surely overdiffusive even though the models are known to have pure point spectrum with exponentially localized eigenstates for almost every configuration of the polymers furthermore the level spacing is shown to be regular at the critical energy

new example of a generating function for wznw fusion rules the authors review the construction of generating functions for wznw wess zumino novikov witten fusion rules and present the explicit form of the generating function for so 5 in addition to providing a compact and manifestly positive expression for the so 5 fusion rules at all levels this result illustrates a general conjecture on the relation between the generating function for wznw fusion rules and that for finite lie algebra tensor products

localization of the continuous anderson hamiltonian in 1 d we study the bottom of the spectrum of the anderson hamiltonian mathcal h l partial x 2 xi on 0 l driven by a white noise xi and endowed with either dirichlet or neumann boundary conditions we show that as l rightarrow infty the point process of the appropriately shifted and rescaled eigenvalues converges to a poisson point process on mathbf r with intensity e x dx and that the appropriately rescaled eigenfunctions converge to dirac masses located at independent and uniformly distributed points furthermore we show that the shape of each eigenfunction recentered around its maximum and properly rescaled is given by the inverse of a hyperbolic cosine we also show that the eigenfunctions decay exponentially from their localization centers at an explicit rate and we obtain very precise information on the zeros and local maxima of these eigenfunctions finally we show that the eigenvalues eigenfunctions in the dirichlet and neumann cases are very close to each other and converge to the same limits

franco american meeting on the mathematics of random and almost periodic potentials we present the abstracts of papers given at a conference held at the ecole polytechnique in june 1983 on the mathematics of random and almost periodic potentials

fine structure of the zeros of orthogonal polynomials iv a priori bounds and clock behavior we prove locally uniform spacing for the zeros of orthogonal polynomials on the real line under weak conditions jacobi parameters approach the free ones and are of bounded variation we prove that for ergodic discrete schrodinger operators poisson behavior implies a positive lyapunov exponent both results depend on a priori bounds on eigenvalue spacings for which we provide several proofs

on the largest singular values of random matrices with independent cauchy entries we apply the method of determinants to study the distribution of the largest singular values of large m n real rectangular random matrices with independent cauchy entries we show that for a special one parametric class of statistics the properties of the largest singular values rescaled by a factor 1 m2n2 agree in the limit with the statistical properties of the poisson random point process with the intensity 1 x 3 2 and therefore are different from the tracy widom law among other corollaries of our method we show an interesting connection between the mathematical expectations of the determinants of the complex rectangular m n standard wishart ensemble and the real rectangular 2m 2n standard wishart ensemble

lacunarity definition for ramified data sets based on optimal cover lacunarity is a measure of how data fills space it complements fractal dimension which measures how much space is filled this paper discusses the limitations of the standard gliding box algorithm for calculating lacunarity which leads to a re examination of what lacunarity is meant to describe two new lacunarity measures for ramified data sets are then presented that more directly measure the gaps in a ramified data set these measures are rigorously defined an algorithm for estimating the new lacunarity measure using fuzzy c means clustering algorithm is developed the lacunarity estimation algorithm is used to analyze two and three dimensional cantor dusts applications for these measures include biological modeling and target detection within ramified data sets

concentration of eigenvalues for skew shift schrodinger operators i analyze the microscopic behavior of the eigenvalues of skew shift schrodinger operators and show that their statistics must resemble the one of the anderson model rather than the one of quasi periodic schrodinger operators

on the joint distribution of energy levels of random schrodinger operators we consider operators with random potentials on graphs such as the lattice version of the random schrodinger operator the main result is a general bound on the probabilities of simultaneous occurrence of eigenvalues in specified distinct intervals with the corresponding eigenfunctions being separately localized within prescribed regions the bound generalizes the wegner estimate on the density of states the analysis proceeds through a new multi parameter spectral averaging principle

generating function method for fusion rules this is the second of two articles devoted to an exposition of the generating function method for computing fusion rules in affine lie algebras the present paper focuses on fusion rules using the machinery developed for tensor products in the companion article although the kac walton algorithm provides a method for constructing a fusion generating function from the corresponding tensor product generating function we describe a more powerful approach which starts by first defining the set of fusion elementary couplings from a natural extension of the set of tensor product elementary couplings a set of inequalities involving the level are derived from this set using farkas lemma these inequalities taken in conjunction with the inequalities defining the tensor products define what we call the fusion basis given this basis the machinery of our previous paper may be applied to construct the fusion generating function new generating functions for sp 4 and su 4 together with a closed form expres

the repulsion between localization centers in the anderson model in this note we show that a simple combination of deep results in the theory of random schrodinger operators yields a quantitative estimate of the fact that the localization centers become far apart as corresponding energies are close together

eigenvalue statistics for lattice hamiltonian with off diagonal disorder this short note deals with a certain kind of lattice hamiltonian with off diagonal disorder based on the exponential decay of the fractional moment of the green function we are able to prove that the properly rescaled eigenvalues of the random hamiltonian are distributed as a poisson point process with intensity measure given by the density of states one of the key step in this proof is the minami type estimate as a crucial ingredient we also use the minami type estimate to study some important properties of the random hamiltonian such as multiplicity of the eigenvalues and quantitative estimate of the localization centers

tracy widom at high temperature we investigate the marginal distribution of the bottom eigenvalues of the stochastic airy operator when the inverse temperature beta tends to 0 we prove that the minimal eigenvalue whose fluctuations are governed by the tracy widom beta law converges weakly when properly centered and scaled to the gumbel distribution more generally we obtain the convergence in law of the marginal distribution of any eigenvalue with given index k those convergences are obtained after a careful analysis of the explosion times process of the riccati diffusion associated to the stochastic airy operator we show that the empirical measure of the explosion times converges weakly to a poisson point process using estimates proved in dumaz and virag ann inst h poincare probab statist 49 4 915 933 2013 we further compute the empirical eigenvalue density of the stochastic airy ensemble on the macroscopic scale when beta rightarrow 0 as an application we investigate the maximal eigenvalues statistics of beta n ensembles when the repulsion parameter beta n rightarrow 0 when n rightarrow infty we study the double scaling limit n rightarrow infty beta n rightarrow 0 and argue with heuristic and numerical arguments that the statistics of the marginal distributions can be deduced following the ideas of edelman and sutton j stat phys 127 6 1121 1165 2007 and ramirez et al j am math soc 24 919 944 2011 from our later study of the stochastic airy operator

on the basic states of one dimensional disordered structures the purpose of this paper is to study a limit probability distribution of the set of the first eigenvalues 1 2 with a fixed and of the boundary problem on the interval 0 wherea t q t are the random stationary processes particularly the question of the repulsion between the first eigenvalues small energetic levels is studied it has been proved that in the divergent case q t 0 a t 0 levels repulsion exists as for the potential case a t 1 q t 0 there is not any repulsion at all this is one of the main differences between these two cases

an efficient implementation of the gliding box lacunarity algorithm lacunarity is a measure of how data fills space it complements fractal dimension which measures how much space is filled many researchers use the gliding box algorithm to calculate lacunarity this paper introduces a fast algorithm for making this calculation this algorithm is akin to fast box counting algorithms used by some researchers to estimate fractal dimension a simplified gliding box measure equation along with key pseudo code implementations for the algorithm are presented applications for the gliding box lacunarity measure have ranged from biological community modelling to target detection

the effect of boundaries on the spectrum of a one dimensional random mass dirac hamiltonian the average density of states dos of the one dimensional dirac hamiltonian with a random mass on a finite interval 0 l is derived our method relies on the eigenvalue distributions extreme value statistics problem which are obtained explicitly the well known dyson singularity is recovered above the crossover energy below c we find a log normal suppression of the average dos

the central limit theorem for the spectrum of the random one dimensional schrodinger operator let hl d2 dt2 q t be an one dimensional random schrodinger operator in 2 l l with the classical boundary conditions the random potential q t has a form q t f xt where xt is a brownian motion on the euclidean v dimensional torus f sv r1 is a smooth function with the nondegenerated critical points minsv f 0 let n l lambda sum lambda i l leqslant lambda 1 lambda i l omega are the eigenvalues of hl be a spectral distribution function in the volume l l and n liml 1 2l nl be a corresponding limit distribution function

lacunarity and universality a new expression for the lacunarity of sierpinski carpets is proposed it turns out that the universality can be described with the new expression the fractal dimension and the connectivity

kronecker products for compact semisimple lie groups a review is given of the application of s function techniques to the evaluation of kronecker products of irreducible representations of compact semisimple lie groups explicit formulae are derived for all irreducible representations of all such groups recent developments involving composite young diagrams are brought to fruition and the vexed problem of so2k is dealt with completely new branching rules for the classical groups are given in an appendix these are exploited in the evaluation of kronecker products by means of a technique which is applied to both classical and exceptional groups a discussion is made of various modification rules which are needed to express the final results in standard form

on the density of states of schrodinger operators with a random potential using very recent results on ergodic theorems for superadditive processes on rd the authors prove the existence of the density of states for a wide class of random schrodinger operators in particular new non asymptotic estimates on the density of states are obtained and examples are discussed

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

correlation estimates in the anderson model we give a new proof of correlation estimates for arbitrary moments of the resolvent of random schrodinger operators on the lattice that generalizes and extends the correlation estimate of minami for the second moment we apply this moment bound to obtain a new n level wegner type estimate that measures eigenvalue correlations through an upper bound on the probability that a local hamiltonian has at least n eigenvalues in a given energy interval another consequence of the correlation estimates is that the results on the poisson statistics of energy level spacing and the simplicity of the eigenvalues in the strong localization regime hold for a wide class of translation invariant selfadjoint lattice operators with decaying off diagonal terms and random potentials

breaking supersymmetry in a one dimensional random hamiltonian the one dimensional supersymmetric random hamiltonian where x is a gaussian white noise of zero mean and variance g presents particular spectral and localization properties at low energy a dyson singularity in the integrated density of states idos n e 1 ln2e and a delocalization transition related to the behavior of the lyapunov exponent inverse localization length vanishing like e 1 ln e as e 0 we study how this picture is affected by breaking supersymmetry with a scalar random potential h hsusy v x where v x is a gaussian white noise of variance in the limit g3 a fraction of states n 0 g ln2 g3 migrate to the negative spectrum and the lyapunov exponent reaches a finite value 0 g ln g3 at e 0 the exponential lifshits tail of the idos for e is studied in detail and is shown to involve a competition between the two noises x and v x whatever the larger is this analysis relies on analytic results for n e and e obtained by two different methods a stochastic method and the replica method the problem of extreme value statistics of eigenvalues is also considered distribution of the nth excited state energy the results are analyzed in the context of classical diffusion in a random force field in the presence of random annihilation creation local rates

eigenvalue statistics for schrodinger operators with random point interactions on rd d 1 2 3 we prove that the local eigenvalue statistics at energy e in the localization regime for schrodinger operators with random point interactions on rd for d 1 2 3 is a poisson point process with the intensity measure given by the density of states at e times the lebesgue measure this is one of the first examples of poisson eigenvalue statistics for the localization regime of multi dimensional random schrodinger operators in the continuum the special structure of resolvent of schrodinger operators with point interactions facilitates the proof of the minami estimate for these models

almost periodic schrodinger operators iii the absolutely continuous spectrum in one dimension we studyh d2 dx2 v x withv x limit periodic e g v x ancos x 2n with an we prove that for a genericv and for generican in the explicit example h is a cantor nowhere dense perfect set for a dense set the spectrum is both cantor and purely absolutely continuous and therefore purely recurrent absolutely continuous

su n and sp 2n wzw fusion rules fusion rules for wzw models based on su n and sp 2n are considered using the results of kac 1989 and walton 1989 1990 it is shown that these fusion rules may be computed using young diagram methods by performing extra modifications these modifications take the form of the removal of boundary strips which bear a striking resemblance to the modifications necessary when performing tensor products for these algebras this fact is exploited to exhibit a duality between the fusion rules of su n at level k and su k at level n and also between sp 2n at level k and so 2k at level n the former duality has been discussed in the context of two dimensional statistical mechanics by kuniba and nakanishi 1990 and also by naculich and schnitzer 1990 in the case of wzw models and goodman and wenzl 1989 for hecke algebras at roots of unity for su 3 a manifestly positive combinatorial procedure for computing fusion rules and a generating function for the fusion coefficients are given

level statistics in a quasiperiodic system the authors examine the level statistics of harper s equation a simple model of a one dimensional quasiperiodic system through the distribution of the spacings between adjacent eigenvalues each normalised by the average local density of states they present exact results for the distribution for both small and large amplitude alpha of the incommensurate potential showing it has a simple form in each of these limits they show numerically that these distributions are preserved as alpha approaches its critical value 2 from each side and also that at the critical point the distribution of the normalised spacings has a distinct but simple form

individual energy level distributions for one dimensional diagonal and off diagonal disorder we study the distribution of the n th energy level for two different one dimensional random potentials this distribution is shown to be related to the distribution of the distance between two consecutive nodes of the wave function we first consider the case of a white noise potential and study the distributions of energy level both in the positive and the negative part of the spectrum it is demonstrated that in the limit of a large system l the distribution of the n th energy level is given by a scaling law which is shown to be related to the extreme value statistics of a set of independent variables in the second part we consider the case of a supersymmetric random hamiltonian potential v x x 2 x we study first the case of x being a white noise with zero mean it is in particular shown that the ground state energy which behaves on average like exp l 1 3 in agreement with previous work is not a self averaging quantity in the limit l as is seen in the case of diagonal disorder then we consider the case when x has a non zero mean value

eigenvalue statistics for random schrodinger operators with non rank one perturbations we prove that certain natural random variables associated with the local eigenvalue statistics for generalized lattice anderson models constructed with finite rank perturbations are compound poisson distributed this distribution is characterized by the fact that the levy measure is supported on at most a finite set of positive integers determined by the rank the proof relies on a minami type estimate for finite rank perturbations for anderson type continuum models on mathbb r d we prove a similar result for certain natural random variables associated with the local eigenvalue statistics we prove that the compound poisson distribution associated with these random variables has a levy measure whose support is at most the set of positive integers