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

localization for the anderson model on a strip with singular potentials abstract we prove exponential localization in the anderson model on a one dimensional strip for any potential whose probability distribution is not concentrated on a single point and possesses some finite moment in particular we obtain localization for bernoulli potentials on a strip

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 note on fractional moments for the one dimensional continuum anderson model we give a proof of dynamical localization in the form of exponential decay of spatial correlations in the time evolution for the one dimensional continuum anderson model via the fractional moments method this follows via exponential decay of fractional moments of the green function which is shown to hold at arbitrary energy and for any single site distribution with bounded compactly supported density

new characterizations of the region of complete localization for random schrodinger operators we study the region of complete localization in a class of random operators which includes random schrodinger operators with anderson type potentials and classical wave operators in random media as well as the anderson tight binding model we establish new characterizations or criteria for this region of complete localization given either by the decay of eigenfunction correlations or by the decay of fermi projections these are necessary and sufficient conditions for the random operator to exhibit complete localization in this energy region using the first type of characterization we prove that in the region of complete localization the random operator has eigenvalues with finite multiplicity

weak disorder localization and lifshitz tails this paper is devoted to the study of localization of discrete random schrodinger hamiltonians in the weak disorder regime consider an i i d anderson model and assume that its left spectral edge is 0 let be the coupling constant measuring the strength of the disorder for small we prove a lifshitz tail type estimate and use it to derive localization in a band starting at 0 going up to a distance of the average of the potential in this energy region we show that the localization length at energy e is bounded from above by a constant times the square root of the distance between e and the average of the potential

constructive fractional moment criteria for localization in random operators abstract we present a family of finite volume criteria which cover the regime of exponential decay for the fractional moments of green functions of operators with random potentials such decay is a technically convenient characterization of localization for it is known to imply spectral localization absence of level repulsion dynamical localization and a related condition which plays a significant role in the quantization of the hall conductance in two dimensional fermi gases the constructive criteria also preclude fast power law decay of the green functions at mobility edges

a lower wegner estimate and bounds on the spectral shift function for continuum random schrodinger operators abstract we prove a strictly positive locally uniform lower bound on the density of states dos of continuum random schrodinger operators on the entire spectrum i e we show that the dos does not have a zero within the spectrum this follows from a lower wegner estimate for finite volume continuum random schrodinger operators we assume throughout iid random variables and the single site distribution having a lebesgue density bounded from below on its support the main mathematical novelty in this paper are pointwise in energy bounds on the expectation of the spectral shift function at all energies for these operators where we mainly focus on perturbations corresponding to a change from dirichlet to neumann boundary conditions along the boundary of a cube we show that the bound scales with the area of the hypersurface where the boundary conditions are changed we also prove bounds on the averaged spectral shift function for perturbations by bounded and compactly supported multiplication operators

eigenvector localization for random band matrices with power law band width it is shown that certain ensembles of random matrices with entries that vanish outside a band around the diagonal satisfy a localization condition on the resolvent which guarantees that eigenvectors have strong overlap with a vanishing fraction of standard basis vectors provided the band width w raised to a power remains smaller than the matrix size n for a gaussian band ensemble with matrix elements given by i i d centered gaussians within a band of width w the estimate 8 holds

a new proof of localization in the anderson tight binding model we give a new proof of exponential localization in the anderson tight binding model which uses many ideas of the frohlich martinelli scoppola and spencer proof but is technically simpler particularly the probabilistic estimates

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

edge detection using two dimensional local structure information abstract local intensity discontinuities commonly referred to as edges are important attributes of an image many imaging scenarios produce image regions exhibiting complex two dimensional 2d local structure such as when several edges meet to form corners and vertices traditional derivative based edge operators which typically assume that an edge can be modeled as a one dimensional 1d piecewise smooth step function give misleading results in such situations leclerc and zucker introduced the concept of local structure as an aid for locating intensity discontinuities they proposed a detailed procedure for detecting discontinuities in a 1d function they had only given a preliminary version of their scheme however for 2d images three related edge detection methods are proposed that draw upon 2d local structural information the first method greatly expands upon leclerc and zucker s 2d method the other two methods employ a mechanism similar to that used by the maximum homogeneity filter a filter used for image enhancement all three methods permit the detection of multiple edges at a point and have the flexibility to detect edges at differing spatial and angular acuity results show that the methods typically perform better than other operators

convergence and stability of a constrained partition based moving horizon estimator a novel type of iterative partition based moving horizon estimators pmhe is proposed the estimates of which approach those of a centralized moving horizon estimator as the number of iterations increases it uses a deterministic setting and can handle known inputs as well as bounds on the estimated state we derive conditions on the system its partitions and the scalar regularization parameter which guarantee convergence towards the optimal centralized state estimate as well as stability of the estimation error dynamics even with a finite number of iterations at each time step finally numerical simulations demonstrate both the features and competitive estimation quality of the proposed method

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

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

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

ground state energy of the one dimensional discrete random schrodinger operator with bernoulli potential in this paper we show the that the ground state energy of the one dimensional discrete random schrodinger operator with bernoulli potential is controlled asymptotically as the system size n goes to infinity by the random variable l n the length the longest consecutive sequence of sites on the lattice with potential equal to zero specifically we will show that for almost every realization of the potential the ground state energy behaves asymptotically as frac pi 2 ell n 1 2 in the sense that the ratio of the quantities goes to one

anderson localization for bernoulli and other singular potentials we prove exponential localization in the anderson model under very weak assumptions on the potential distribution in one dimension we allow any measure which is not concentrated on a single point and possesses some finite moment in particular this solves the longstanding problem of localization for bernoulli potentials i e potentials that take only two values in dimensions greater than one we prove localization at high disorder for potentials with holder continuous distributions and for bounded potentials whose distribution is a convex combination of a holder continuous distribution with high disorder and an arbitrary distribution these include potentials with singular distributions we also show that for certain bernoulli potentials in one dimension the integrated density of states has a nontrivial singular component

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

asymptotic ergodicity of the eigenvalues of random operators in the localized phase we prove that for a general class of random operators the family of the unfolded eigenvalues in the localization region is asymptotically ergodic in the sense of minami spectra of random operators and related topics 2011 minami conjectured this to be the case for discrete anderson model in the localized regime we also provide a local analogue of this result from the asymptotics ergodicity one can recover the statistics of the level spacings as well as a number of other spectral statistics our proofs rely on the analysis developed in germinet and klopp spectral statistics for random schrodinger operators in the localized regime 2010

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

anderson localization for multi dimensional systems at large disorder or large energy we prove that discrete schrodinger operators on d with a random potential have almost surely only pure point spectrum and exponentially decaying eigenfunctions for large disorder or large energy this is the first proof of localization for multi dimensional anderson models

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

localization at large disorder and at extreme energies an elementary derivation the work presents a short proof of localization under the conditions of either strong disorder 0 or extreme energies for a wide class of self adjoint operators with random matrix elements acting inl 2 spaces a prototypical example is the discrete schrodinger operatorh u 0 x v x onz d d 1 withu 0 x a specified background potential and v x generated as random variables the general results apply to operators with replaced by a non local self adjoint operatort whose matrix elements satisfy y t x y s const uniformly inx for somes 1 localization means here that within a specified energy range the spectrum ofh is of the pure point type or equivalently the wave functions do not spread indefinitely under the unitary time evolution generated byh the effect is produced by strong disorder in either the potential or in the off diagonal matrix elementst x y under rapid decay oft x y the corresponding eigenfunctions are also proven to decay exponentially the method is based on resolvent techniques the central technical ideas include the use of low moments of the resolvent kernel i e g e x y s withs small enough 1 to avoid the divergence caused by the distribution s cauchy tails and an effective use of the simple form of the dependence ofg e x y on the individual matrix elements ofh in elucidating the implications of the fundamental equation h e g e x x 0 x x0 this approach simplifies previous derivations of localization results avoiding the small denominator difficulties which have been hitherto encountered in the subject it also yields some new results which include localization under the following sets of conditions i potentials with an inhomogeneous non random partu 0 x ii the bethe lattice iii operators with very slow decay in the off diagonal terms t x y 1 x y d e and iv localization produced by disordered boundary conditions

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

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

the local structure of image discontinuities in one dimension the detailed structure of intensities in the local neighborhood of an edge can often indicate the nature of the physical event givinig rise to that edge we argue that the limit as we approach arbitrarily close to either side of an edge of such image parameters as type of texture texture gradient color appropriate directional derivatives of intensity etc is a key aspect of this structure however the general problem of capturing this local structure is surprisingly complex thus we restrict ourselves in this paper to a relatively simple domain one dimensional cuts through idealized images modeled by piecewise smooth c1 functions corrupted by gaussian noise within this domain we define local structure to be the limit of the uncorrupted intensity and of its derivatives as we approach arbitrarily close to either side of a discontinuity we develop a technique that captures this local structure while simultaneously locating the discontinuities and demonstrate that these tasks are in fact inseparable the technique is an extension using estimation theory of the classical definition of discontinuity it handles in a consistent fashion both jump discontinuities in the function and jump discontinuities in its first derivative so called step edges are a special case of the former and roof edges of the latter it also integrates again in a consistent fashion information derived from a number of different neighborhood sizes

intermixture of extended edge and localized bulk energy levels in macroscopic hall systems we study the spectrum of a random schrodinger operator for an electron subjected to a magnetic field in a finite but macroscopic two dimensional system of linear dimensions equal to l the y direction is periodic and in the x direction the electron is confined by two smooth increasing boundary potentials the eigenvalues of the hamiltonian are classified according to their associated quantum mechanical diamagnetic current in the y direction here we look at an interval of energies inside the first landau band of the random operator for the infinite plane in this energy interval with large probability there exist l eigenvalues with positive or negative currents of 1 between each of these there exist l2 eigenvalues with infinitesimal current e b log l 2 we explain the relevance of this analysis of boundary diamagnetic currents to the integer quantum hall effect

finite volume fractional moment criteria for anderson localization a technically convenient signature of localization exhibited by discrete operators with random potentials is exponential decay of the fractional moments of the green function within the appropriate energy ranges known implications include spectral localization absence of level repulsion strong form of dynamical localization and a related condition which plays a significant role in the quantization of the hall conductance in two dimensional fermi gases we present a family of finite volume criteria which under some mild restrictions on the distribution of the potential cover the regime where the fractional moment decay condition holds the constructive criteria permit to establish this condition at spectral band edges provided there are sufficient lifshitz tail estimates on the density of states they are also used here to conclude that the fractional moment condition and thus the other manifestations of localization are valid throughout the regime covered by the multiscale analysis in the converse direction the analysis rules out fast power law decay of the green functions at mobility edges

extremal theory for spectrum of random discrete schrodinger operator iii localization properties in this paper we study the asymptotic localization properties with high probability of the kth eigenfunction associated with the kth largest eigenvalue k 1 fixed of the multidimensional anderson hamiltonian in torus v increasing to the whole of lattice denote by z k v v the site at which the kth largest value of potential is attained it is well known that if the tails of potential distribution are heavier than the double exponential function and satisfies additional regularity and continuity conditions at infinity then the kth eigenfunction is asymptotically delta function at the site z k v localization centre for some random k v k 1 we study the asymptotic behavior of the index v k by distinguishing between three cases of the tails of potential distribution i for the heavy tails including gaussian v k is asymptotically bounded ii for the light tails but heavier than the double exponential the index v k unboundedly increases like v o 1 iii finally for the double exponential tails with high disorder the index v k behaves like a power of v for weibull s and fractional double exponential types distributions associated with the case ii we obtain the first order expansion formulas for log v k

a central limit theorem for products of random matrices and goe statistics for the anderson model on long boxes we consider products of random matrices that are small independent identically distributed perturbations of a fixed matrix mathcal t 0 focusing on the eigenvalues of mathcal t 0 of a particular size we obtain a limit to a sde in a critical scaling previous results required mathcal t 0 to be a conjugated unitary matrix so it could not have eigenvalues of different modulus from the result we can also obtain a limit sde for the markov process given by the action of the random products on the flag manifold applying the result to random schrodinger operators we can improve some results by valko and virag showing goe statistics for the rescaled eigenvalue process of a sequence of anderson models on long boxes in particular we solve a problem posed in their work

an eigensystem approach to anderson localization abstract we introduce a new approach for proving localization pure point spectrum with exponentially decaying eigenfunctions dynamical localization for the anderson model at high disorder in contrast to the usual strategy we do not study finite volume green s functions instead we perform a multiscale analysis based on finite volume eigensystems eigenvalues and eigenfunctions information about eigensystems at a given scale is used to derive information about eigensystems at larger scales this eigensystem multiscale analysis treats all energies of the finite volume operator at the same time establishing level spacing and localization of eigenfunctions in a fixed box with high probability a new feature is the labeling of the eigenvalues and eigenfunctions by the sites of the box

the equations for the moduli space of n points on the line a central question in invariant theory is that of determining the relations among invariants geometric invariant theory git quotients come with a natural ample line bundle and hence often a natural projective embedding this question translates to determining the equations of the moduli space under this embedding this article deals with one of the most classical quotients the space of ordered points on the projective line we show that under any weighting of the points this quotient is cut out scheme theoretically by a particularly simple set of quadric relations with the single exception of the segre cubic threefold the space of six points with equal weight we also show that the ideal of relations is generated in degree at most four and we give an explicit description of the generators if all the weights are even e g in the case of equal weight for odd n then we show that the ideal of relations is generated by quadrics

uniform anderson localization unimodal eigenstates and simple spectra in a class of haarsh deterministic potentials abstract we consider a particular class of families of multi dimensional lattice schrodinger operators with deterministic e g quasi periodic potentials generated by the hull given by an orthogonal series over the haar wavelet basis on the torus of arbitrary dimension with expansion coefficients considered as independent parameters in the strong disorder regime we prove anderson localization for generic operator families and show that all localized eigenfunctions are unimodal and feature uniform exponential decay away from their respective localization centers we also prove a variant of the minami estimate for deterministic potentials and simplicity of the spectrum

finite volume approximation of the anderson model in the anderson model on zd we consider a sequence of its finite volume approximation hk k and construct a set of sequences composed of the eigenvalues and eigenfunctions of hk in the localized region i which converge to those of h simultaneously for its proof minami s estimate turns out to be important this result implies that in the localized region each eigenfunction behaves almost independently around their centers of localization

quantum harmonic oscillator systems with disorder we study many body properties of quantum harmonic oscillator lattices with disorder a sufficient condition for dynamical localization expressed as a zero velocity lieb robinson bound is formulated in terms of the decay of the eigenfunction correlators for an effective one particle hamiltonian we show how state of the art techniques for proving anderson localization can be used to prove that these properties hold in a number of standard models we also derive bounds on the static and dynamic correlation functions at both zero and positive temperature in terms of one particle eigenfunction correlators in particular we show that static correlations decay exponentially fast if the corresponding effective one particle hamiltonian exhibits localization at low energies regardless of whether there is a gap in the spectrum above the ground state or not our results apply to finite as well as to infinite oscillator systems the eigenfunction correlators that appear are more general than those previously studied in the literature in particular we must allow for functions of the hamiltonian that have a singularity at the bottom of the spectrum we prove exponential bounds for such correlators for some of the standard models

some defects in finite difference edge finders this work illustrates and explains various artifacts in the output of five finite difference edge finders those of j f canny 1983 1986 r a boie et al 1986 and r a boie and i j cox 1987 and three variations on that of d marr and e c hildreth 1980 reimplemented with a common output format and method of noise suppression these artifacts include gaps in boundaries spurious boundaries and deformation of region shape

decorrelation estimates for some continuous and discrete random schrodinger operators in dimension one and applications to spectral statistics abstract the purpose of the present work is to establish decorrelation estimates at distinct energies for some random schrodinger operator in dimension one in particular we establish the result for some random operators on the continuum with alloy type potential these results are used to give a description of the spectral statistics

on mott s formula for the ac conductivity in the anderson model we study the ac conductivity in linear response theory in the general framework of ergodic magnetic schrodinger operators for the anderson model if the fermi energy lies in the localization regime we prove that the ac conductivity is bounded from above by c 2 log 1 d 2 at small frequencies this is to be compared to mott s formula which predicts the leading term to be c 2 log 1 d 1

anderson localization and the space time characteristic of continuum states a proof of anderson localization is obtained by ruling out any continuous spectrum on the basis of the space time characteristic of its states

mott law as upper bound for a random walk in a random environment we consider a random walk on the support of an ergodic simple point process on mathbb r d d 2 furnished with independent energy marks the jump rates of the random walk decay exponentially in the jump length and depend on the energy marks via a boltzmann type factor this is an effective model for the phonon induced hopping of electrons in disordered solids in the regime of strong anderson localization under some technical assumption on the point process we prove an upper bound for the diffusion matrix of the random walk in agreement with mott law a lower bound for d 2 in agreement with mott law was proved in 8

simplicity of eigenvalues in the anderson model we give a transparent and intuitive proof that all eigenvalues of the anderson model in the region of localization are simple

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

generalized eigenvalue counting estimates for the anderson model we generalize minami s estimate for the anderson model and its extensions to n eigenvalues allowing for n arbitrary intervals and arbitrary single site probability measures with no atoms as an application we derive new results about the multiplicity of eigenvalues and mott s formula for the ac conductivity when the single site probability distribution is holder continuous

inverse tunnelingestimates and applications to the study of spectral statistics ofrandom operators on the real line we present a proof of a minami type estimate for one dimensional random schrodinger operators valid at all energies in the localization regime provided a wegner estimate is known to hold the minami type estimate is then applied to two models to obtain results on their spectral statistics the heuristics underlying our proof of minami type estimates is that close by eigenvalues of a one dimensional schrodinger operator correspond either to eigenfunctions that live far away from each other in space or they come from some tunneling phenomena in the second case one can undo the tunneling and thus construct quasi modes that live far away from each other in space resume nous demontrons une inegalite de type minami pour des operateurs de schrodinger aleatoires uni dimensionnel dans toute la region localisee si une estimee de wegner est connue cette estimee de type de minami est alors appliquee pour obtenir les statistiques spectrales pour deux modeles l heuristique qui guide ce travail est que des valeurs propres proches pour un operateur de schrodinger sur un intervalle sont soient localisees loin l une de l autre soit sont la consequence d un phenomene d effet tunnel dans le second cas on peut en defaisant cet effet tunnel construire des quasi modes qui sont localises loin l un de l autre 0 introduction consider the following two random operators on the real line the anderson model 0 1 h d2 dx2 w

extremal theory for spectrum of random discrete schrodinger operator ii distributions with heavy tails we study the asymptotic structure of the first k largest eigenvalues k v and the corresponding eigenfunctions k v of a finite volume anderson model discrete schrodinger operator mathcal h v kappa delta v xi cdot on the multidimensional lattice torus v increasing to the whole of lattice provided the distribution function f of i i d potential satisfies condition log 1 f t o t3 and some additional regularity conditions as t for z v denote by 0 z the principal eigenvalue of the single peak hamiltonian v z z in l2 v and let lambda 0 k v be the kth largest value of the sample 0 in v we first show that the eigenvalues k v are asymptotically close to lambda 0 k v we then prove extremal type limit theorems i e poisson statistics for the normalized eigenvalues k v bv av where the normalizing constants av 0 and bv are chosen the same as in the corresponding limit theorems for lambda 0 k v the eigenfunction k v is shown to be asymptotically completely localized as v at the sites zk v v defined by lambda 0 z k v lambda 0 k v proofs are based on the finite rank in particular rank one perturbation arguments for discrete schrodinger operator when potential peaks are sparse

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

level spacing for non monotone anderson models we prove localization and probabilistic bounds on the minimum level spacing for a random block anderson model without monotonicity using a sequence of narrowing energy windows and associated schur complements we obtain detailed probabilistic information about the microscopic structure of energy levels of the hamiltonian as well as the support and decay of eigenfunctions

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