lyapunov exponents for unitary anderson models we study a unitary version of the one dimensional anderson model given by a five diagonal deterministic unitary operator multiplicatively perturbed by a random phase matrix we fully characterize positivity and vanishing of the lyapunov exponent for this model throughout the spectrum and for arbitrary distributions of the random phases this includes bernoulli distributions where in certain cases a finite number of critical spectral values with vanishing lyapunov exponent exist we establish similar results for a unitary version of the random dimer model

upper bounds on wavepacket spreading for random jacobi matrices a method is presented for proving upper bounds on the moments of the position operator when the dynamics of quantum wavepackets is governed by a random possibly correlated jacobi matrix as an application one obtains sharp upper bounds on the diffusion exponents for random polymer models coinciding with the lower bounds obtained in a prior work the second application is an elementary argument not using multiscale analysis or the aizenman molchanov method showing that under the condition of uniformly positive lyapunov exponents the moments of the position operator grow at most logarithmically in time

absence of localization in disordered systems with local correlation in an early work by dunlap et al 1990 it was conjectured using a matrix transfer approach that diagonal disordered systems with local correlation dimer models exhibit an absence of anderson localization in this letter we find explicitly these delocalized states and energies using a suitable unitary transformation and the symmetries related to the model moreover we expand these states around the critical energy and find a new 1 square root e ec divergence

low mean internodal distance network topologies and simulated annealing networks have been found through simulated annealing with mean internodal distances lower than any previously reported for a given number of nodes n with a maximum of p outgoing links per node these networks form the closest known approximations to moore networks however the improvements in mean internodal distance obtained are relatively small a few percent and this improvement decreases rapidly with increasing p the relation between average nodal switching complexity and mean internodal distance is briefly explored through simulated annealing initial sharp decreases in mean internodal distance can be obtained through moderate increases in switching complexity however further decreases require much larger increases in average complexity

dynamical localization for the random dimer schrodinger operator we study the one dimensional random dimer model with hamiltonian h v where for all x mathbb z v 2x v 2x 1 and where the v 2x are i i d bernoulli random variables taking the values v v 0 we show that for all values of vand with probability one in the spectrum of his pure point if v 1 and v 1 sqrt 2 the lyapunov exponent vanishes only at the two critical energies given by e v for the particular value v 1 sqrt 2 respectively v sqrt 2 we show the existence of new additional critical energies at e 3 sqrt 2 respectively e 0 on any compact interval inot containing the critical energies the eigenfunctions are then shown to be semi uniformly exponentially localized and this implies dynamical localization for all q 0 and for all ell 2 mathbb z with sufficiently rapid decrease mathop sup limits t r psi i left q right kern 1pt left t right mathop sup limits t left langle p i left h omega right psi t left x right q p i left h omega right psi t right rangle infty here psi t e ih omega t psi and pi h is the spectral projector of h onto the interval i in particular if v 1 and v sqrt 2 these results hold on the entire spectrum so that one can take i h

spectra and transport in almost periodic dimers we study spectral properties of discrete schrodinger operators with potentials obtained via dimerization of a class of aperiodic sequences it is shown that both the nature of the autocorrelation measure of a regular sequence and the presence of generic full probability singular continuous spectrum in the hull of primitive and palindromic four block substitution potentials are robust under dimerization generic results also hold for circle potentials we illustrate these results with numerical studies of the quantum mean square displacement as a function of time the numerical techniques provide a very fast algorithm for the time evolution of wave packets

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

a characterization of the anderson metal insulator transport transition we investigate the anderson metal insulator transition for random schrodinger operators we define the strong insulator region to be the part of the spectrum where the random operator exhibits strong dynamical localization in the hilbert schmidt norm we introduce a local transport exponent e and set the weak metallic transport region to be the part of the spectrum with nontrivial transport i e e 0 we prove that these insulator and metallic regions are complementary sets in the spectrum of the random operator and that the local transport exponent e provides a characterization of the metal insulator transport transition moreover we show that if there is such a transition then e has to be discontinuous at a transport mobility edge more precisely we show that if the transport is nontrivial then e 1 2d where d is the space dimension these results follow from a proof that slow transport of quantum waves in random media implies the starting hypothesis for the authors bootstrap multiscale analysis we also conclude that the strong insulator region coincides with the part of the spectrum where we can perform a bootstrap multiscale analysis proving that the multiscale analysis is valid all the way up to a transport mobility edge

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

an extension of kotani s theorem to random generalized sturm liouville operators we consider the random operator d m dx d dx q x wherem dx andq x are a stationary ergodic random measure and a random function respectively to this general case we extend kotani s theorem which asserts that the absolutely continuous spectrum is completely determined by the ljapounov indices our framework includes the case of stochastic jacobi matrices treated by simon

mean internodal distance in regular and random multihop networks the minimum necessary aggregate link capacity in a telecommunication network is directly proportional to the mean distance between nodes the mean internodal distance is therefore an important network characteristic it is shown that most network topologies including those constructed at random display mean internodal distances comparable to those of many carefully designed networks thus careful selection of network topology to minimize the mean internodal distance may be important in only the most sensitive applications furthermore even in such sensitive applications an almost randomly chosen network topology may be the best choice

perturbation theory for the random dimer model the authors study a one dimensional tight binding model with a random potential taking two values or nu with the restriction that on pairs of two neighbouring sites the potentials take the same value this random dimer model proposed by dunlap phillips and wu 1990 has a vanishing lyapunov exponent for the energy e or nu they compute the lyapunov exponent and the density of states perturbatively in the vicinity of this energy using the invariant measure formalism

localization for a class of one dimensional quasi periodic schrodinger operators we prove for small and satisfying a certain diophantine condition the operator h varepsilon 2 delta frac 1 2 pi cos 2 pi j alpha theta j in mathbb z has pure point spectrum for almost all a similar result is established at low energy for h frac d 2 dx 2 k 2 cos 2 pi x cos 2 pi alpha x theta providedk is sufficiently large

dynamical delocalization in random landau hamiltonians we prove the existence of dynamical delocalization for random landau hamiltonians near each landau level since typically there is dynamical localization at the edges of each disordered broadened landau band this implies the existence of at least one dynamical mobility edge at each landau band namely a boundary point between the localization and delocalization regimes which we prove to converge to the corresponding landau level as either the magnetic field goes to infinity or the disorder goes to zero in this article we prove the existence of dynamical delocalization for random landau hamiltonians near each landau level more precisely we prove that for these two dimensional hamiltonians there exists at least one energy e near each landau level such that e 1 where e the local transport exponent introduced in gk5 is a measure of the rate of transport in wave packets with spectral support near e since typically there is dynamical localization at the edges of each disordered broadened landau band this implies the existence of at least one dynamical mobility edge at each landau band namely a boundary point between the localization and delocalization regimes which we prove to converge to the corresponding landau level as either the magnetic field goes to infinity or the disorder goes to zero random landau hamiltonians are the subject of intensive study due to their links with the integer quantum hall e ect kli for which von klitzing received the 1985 nobel prize in physics they describe an electron moving in a very thin flat conductor with impurities under the influence of a constant magnetic field perpendicular to the plane of the conductor and play an important role in the understanding of the quantum hall e ect l aoa t h nt ku be avss bees laughlin s argument l as pointed out by halperin h uses the assumption that under weak disorder and strong magnetic field the energy spectrum consists of bands of extended states separated

landau hamiltonians with random potentials localization and the density of states we prove the existence of localized states at the edges of the bands for the two dimensional landau hamiltonian with a random potential of arbitrary disorder provided that the magnetic field is sufficiently large the corresponding eigenfunctions decay exponentially with the magnetic field and distance we also prove that the integrated density of states is lipschitz continuous away from the landau energies the proof relies on a wegner estimate for the finite area magnetic hamiltonians with random potentials and exponential decay estimates for the finitearea green s functions the proof of the decay estimates for the green s functions uses fundamental results from two dimensional bond percolation theory

critical energies in random palindrome models we investigate the occurrence of critical energies where the lyapunov exponent vanishes in random schrodinger operators when the potentials have some local order which we call random palindrome models we give necessary and sufficient conditions for the presence of such critical energies the commutativity of finite word elliptic transfer matrices finally we perform some numerical calculations of the lyapunov exponents showing their behavior near the critical energies and the respective time evolution of an initially localized wave packet obtaining the exponent ruling the algebraic growth of the second momentum we also consider special random palindrome models with one letter bounded gap property the transport effects of such long range order are showed numerically

anderson localization for the almost mathieu equation a nonperturbative proof we prove that for any diophantine rotation angle and a e phase the almost mathieu operator h n n 1 n 1 cos 2 n n has pure point spectrum with exponentially decaying eigenfunctions for 15 we also prove the existence of some pure point spectrum for any 5 4

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

localization for random and quasiperiodic potentials a survey is made of some recent mathematical results and techniques for schrodinger operators with random and quasiperiodic potentials a new proof of localization for random potentials established in collaboration with h von dreifus is sketched

dynamical localization for discrete and continuous random schrodinger operators we show for a large class of random schrodinger operators h on and on that dynamical localization holds i e that with probability one for a suitable energy interval i and for q a positive real here is a function of sufficiently rapid decrease and p i h is the spectral projector of h corresponding to the interval i the result is obtained through the control of the decay of the eigenfunctions of h and covers in the discrete case the anderson tight binding model with bernoulli potential dimension 1 or singular potential 1 and in the continuous case anderson as well as random landau hamiltonians

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

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

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

singular unbounded random jacobi matrices there have been several recent proofs of one dimensional anderson localization based on positive lyapunov exponent that hold for bounded potentials we provide a lyapunov exponent based proof for unbounded potentials simultaneously treating the singular and unbounded jacobi case by extending the techniques in a recent work by jitomirskaya and zhu there have been several recent proofs of one dimensional anderson localization based on positive lyapunov exponent that hold for bounded potentials we provide a lyapunov exponent based proof for unbounded potentials simultaneously treating the singular and unbounded jacobi case by extending the techniques in a recent work by jitomirskaya and zhu

absence of diffusion in the anderson tight binding model for large disorder or low energy we prove that the green s function of the anderson tight binding hamiltonian decays exponentially fast at long distances on v with probability 1 we must assume that either the disorder is large or the energy is sufficiently low our proof is based on perturbation theory about an infinite sequence of block hamiltonians and is related to kam methods

singular continuous spectrum under rank one perturbations and localization for random hamiltonians we consider a selfadjoint operator a and a selfadjoint rank one projection p onto a vector which is cyclic for a in terms of the spectral measure d a we give necessary and sufficient conditions for a p to have empty singular continuous spectrum or to have only point spectrum for a e we apply these results to questions of localization in the one and multi dimensional anderson models

upper bounds in quantum dynamics of course the case of main interest is where h is given by l2 rd or 2 zd h is a schr dinger operator of the form a v and ip 0 is a localized wavepacket under the time evolution 1 the wavepacket i t will in general spread out with time finding quantitative bounds for this spreading in concrete cases and developing methods for proving such bounds has been the objective of a great number of papers in recent years one is often interested in the growth behavior of the moments of the position operator that is the function 11 x p 2 i 2 these questions are particularly complicated and interesting in situations where singular continuous spectra occur the discovery of the occurrence of such spectra for many models of physical relevance has triggered some activity on the quantum dynamical side for models with absolutely continuous spectra on the other hand quantum dynamical questions have been investigated much earlier in the context of scattering theory and have been answered to a great degree of satisfaction models with point spectrum or more restrictively models exhibiting anderson localization have also been investigated from a quantum dynamical point of view while it is not true in general that spectral localization implies dynamical localization 26 73 it has been shown that the three main approaches to spectral localization the fractional moment method 3 4 5 multi scale analysis 28 27 75 and the bourgain goldstein schlag method 12 37 yield dynamical localization as a by product 2 13 22 31 34 quantum dynamics for schr dinger operators with singular continuous spectra is much less understood the two primary examples motivated by physics are the harper model also known as the almost mathieu operator at critical cou pling which describes a bloch electron in a constant magnetic field with irrational flux through a unit cell and the fibonacci model which describes a standard one dimensional quasicrystal both models have been studied heavily in both the physics and mathematics communities

localization for off diagonal disorder and for continuous schrodinger operators we extend the proof of localization by delyon levy and souillard to accommodate the anderson model with off diagonal disorder and the continuous schrodinger equation with a random potential

proof of dynamical localization for perturbations of discrete 1d schrodinger operators with uniform electric fields we present a proof of discrete spectrum and dynamical localization for small perturbations of discrete one dimensional schrodinger operators with uniform electric fields the proof of dynamical localization is based on the kam technique

localization for one dimensional continuum bernoulli anderson models we use scattering theoretic methods to prove strong dynamical and exponential localization for one dimensional continuum anderson type models with singular distributions in particular the case of a bernoulli dis tribution is covered the operators we consider model alloys composed of at least two distinct types of randomly dispersed atoms our main tools are the reflection and transmission coefficients for compactly supported single site perturbations of a periodic background which we use to verify the necessary hypotheses of multi scale analysis we show that non reflectionless single sites lead to a discrete set of exceptional energies away from which localization occurs

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

kotani theory for one dimensional stochastic jacobi matrices we consider families of operators h on l2 given by h u n u n 1 u n 1 v n u n wherev is a stationary bounded ergodic sequence we prove analogs of kotani s results including that for a e ac h is the essential closure of the set ofe where e the lyaponov index vanishes and the result that ifv is non deterministic then ac is empty

dynamical localization ii with an application to the almost mathieu operator several recent works have established dynamical localization for schrodinger operators starting from control on the localization length of their eigenfunctions in terms of their centers of localization we provide an alternative way to obtain dynamical localization without resorting to such a strong condition on the exponential decay of the eigenfunctions furthermore we illustrate our purpose with the almost mathieu operator h cos 2 x 15 and with good diophantine properties more precisely for almost all for all q 0 and for all functions l2 mathbb z of compact support we show that mathop sup limits t left langle e ith theta lambda omega psi left x right q e ith theta lambda omega psi right rangle c psi the proof applies equally well to discrete and continuous random hamiltonians in all cases it uses as input a repulsion principle of singular boxes supplied in the random case by the multi scale analysis

dynamical delocalization for the 1d bernoulli discrete dirac operator a 1d tight binding version of the dirac equation is considered after checking that it recovers the usual discrete schrodinger equation in the nonrelativistic limit it is found that for two valued bernoulli potentials the zero mass case presents the absence of dynamical localization for some specific values of the energy albeit it has no continuous spectrum for the other energy values again excluding some very specific ones the bernoulli dirac system is localized independently of the mass

mixed lower bounds for quantum transport let h be a self adjoint operator on a separable hilbert space h h 1 given an orthonormal basis b en of h we consider the time averaged moments x p t of the position operator associated to b we derive lower bounds for the moments in terms of both spectral measure and generalized eigenfunctions u n x of the state as a particular corollary we generalize the recently obtained lower bound in terms of multifractal dimensions of and give some equivalent forms of it which can be useful in applications we establish in particular the relations between the lq norms q 1 2 of the imaginary part of borel transform of probability measures and the corresponding multifractal dimensions

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

lower transport bounds for one dimensional continuum schrodinger operators we prove quantum dynamical lower bounds for one dimensional continuum schrodinger operators that possess critical energies for which there is slow growth of transfer matrix norms and a large class of compactly supported initial states this general result is applied to a number of models including the bernoulli anderson model with a constant single site potential

quantum hamiltonians with quasi ballistic dynamics and point spectrum abstract consider the family of schrodinger operators and also its dirac version on l 2 z or l 2 n h s w f s n w where s is a transformation on compact metric f is a real lipschitz function and w is a sufficiently fast power decaying perturbation under certain conditions it is shown that h s w presents quasi ballistic dynamics for in a dense g set applications include potentials generated by rotations of the torus with analytic condition on f doubling map axiom a dynamical systems and the anderson model if w is a rank one perturbation examples of h s w with quasi ballistic dynamics and point spectrum are also presented

anderson localization for the almost mathieu equation ii point spectrum for lambda 2 we prove that for any 2 and a e the pure point spectrum of the almost mathieu operator h n n 1 n 1 cos 2 n n contains the essential closure hat sigma of the spectrum corresponding eigenfunctions decay exponentially the singular continuous component if it exists is concentrated on a set of zero measure which is nowhere dense in hat sigma

constructive proof of localization in the anderson tight binding model we prove that for large disorder or near the band tails the spectrum of the anderson tight binding hamiltonian with diagonal disorder consists exclusively of discrete eigenvalues the corresponding eigenfunctions are exponentially well localized these results hold in arbitrary dimension and with probability one in one dimension we recover the result that all states are localized for arbitrary energies and arbitrarily small disorder our techniques extend to other physical systems which exhibit localization phenomena such as infinite systems of coupled harmonic oscillators or random schrodinger operators in the continuum

localization for discrete one dimensional random word models we consider schrodinger operators in l2 z whose potentials are obtained by randomly concatenating words from an underlying set w according to some probability measure on w our assumptions allow us to consider models with local correlations such as the random dimer model or more generally random polymer models we prove spectral localization and away from a finite set of exceptional energies dynamical localization for such models these results are obtained by employing scattering theoretic methods together with furstenberg s theorem to verify the necessary input to perform a multiscale analysis

on the problem of dynamical localization in the nonlinear schrodinger equation with a random potential we prove a dynamical localization in the nonlinear schrodinger equation with a random potential for times of the order of o 2 w here is the strength of the nonlinear ity

metal insulator transition for the almost mathieu operator we prove that for diophantine and almost every the almost mathieu operator h n n 1 ni 1 cos 2 n n exhibits localization for 2 and purely absolutely continuous spectrum for 2

localization for random schrodinger operators with correlated potentials we prove localization at high disorder or low energy for lattice schrodinger operators with random potentials whose values at different lattice sites are correlated over large distances the class of admissible random potentials for our multiscale analysis includes potentials with a stationary gaussian distribution whose covariance functionc x y decays as x y where 0 can be arbitrarily small and potentials whose probability distribution is a completely analytical gibbs measure the result for gaussian potentials depends on a multivariable form of nelson s best possible hypercontractive estimate

bounded sobolev norms for linear schrodinger equations under resonant perturbations abstract we prove that the sobolev norms for a 1 d periodic schrodinger equation remain bounded under small resonant perturbations

spectral analysis of the disordered stochastic 1 d ising model we consider the generator of the glauber dynamics for a 1 d ising model with random bounded potential at any temperature we prove that for any realization of the potential the spectrum of the generator is the union of separate branches so called k particles branches k 0 1 2 and with probability one it is a nonrandom set we find the location of the spectrum and prove the localization for the one particle branch of the spectrum

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

localization in one dimensional random media a scattering theoretic approach we use scattering theoretic methods to prove exponential localization for random displacement models in one dimension the operators we consider model both quantum and classical wave propagation our main tools are the reflection and transmission coefficients for compactly supported single site perturbations we show that randomly displaced non reflectionless single sites lead to localization

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

how to prove dynamical localization let h be a self adjoint operator on l 2 z d or l 2 r d with pure point spectrum on some interval i we establish general necessary and sufficient conditions for dynamical localization for a given vector and on the interval of energies i the sufficient conditions we obtain improve the existing ones such as sule or wule and can be useful in applications

anderson localization for the almost mathieu equation iii semi uniform localization continuity of gaps and measure of the spectrum we show that the almost mathieu operator has semi uniform and thus dynamical localization for 15 and a e we also obtain a new estimate on gap continuity in for this operator with 29 or 4 29 and use it to prove that the measure of its spectrum is equal to for in this range and all irrational s

anderson localization for one and quasi one dimensional systems we prove almost sure exponential localization of all the eigenfunctions and nondegeneracy of the spectrum for random discrete schrodinger operators on one and quasi one dimensional lattices this paper provides a much simpler proof of these results than previous approaches and extends to a much wider class of systems we remark in particular that the singular continuous spectrum observed in some quasiperiodic systems disappears under arbitrarily small local perturbations of the potential our results allow us to prove that e g for strong disorder the smallest positive lyapunov exponent of some products of random matrices does not vanish as the size of the matrices increases to infinity

localization in general one dimensional random systems we discuss two ways of extending the recent ideas of localization from discrete schrodinger operators jacobi matrices to the continuum case one case allows us to prove localization in the goldshade molchanov pastur model for a larger class of functions than previously the other method studies the model v wherev is a random constant in each hyper cube we extend wegner s result on the lipschitz nature of the ids to this model

from fixed energy localization analysis to dynamical localization an elementary path we review several techniques initiated by a remarkable work written by spencer j stat phys 51 1009 1019 1988 a quarter of century ago used and further developed in numerous subsequent researches we also describe a fairly general elementary derivation of spectral and strong dynamical anderson localization from the fixed energy analysis of the green s functions on locally finite graphs of polynomial growth obtained either by the multi scale analysis or by the fractional moment method this derivation goes in the same direction as the simon wolf method commun pure appl math 39 75 90 1986 but provides more quantitative estimates can be adapted to multi particle models and combined with a simplified variant of the germinet klein argument commun math phys 222 415 448 2001 results in an elementary proof of strong dynamical localization on arbitrary graphs of polynomial growth

edge and impurity effects on quantization of hall currents we consider the edge hall conductance and show it is invariant under perturbations located in a strip along the edge decaying perturbations far from the edge are also allowed this enables us to prove for the edge conductances a general sum rule relating currents due to the presence of two different media located respectively on the left and on the right half plane as a particular interesting case we put forward a general quantization formula for the difference of edge hall conductances in semi infinite samples with and without a confining wall it implies in particular that the edge hall conductance takes its ideal quantized value under a gap condition for the bulk hamiltonian or under some localization properties for a random bulk hamiltonian provided one first regularizes the conductance we shall discuss this regularization issue our quantization formula also shows that deviations from the ideal value occurs if a semi infinite distribution of impurity potentials is repulsive enough to produce current carrying surface states on its boundary

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 absence of diffusion near the bottom of the spectrum for a random schrodinger operator on l 2 bf r nu we consider a random schrodinger operator onl2 v of the form h omega delta v omega v omega x sigma chi c i x q i omega ci being a covering of v with unit cubes around the sites of v and qi i i d random variables with values in 0 1 we assume that theqi s are continuously distributed with bounded densityf q and that 0 p q0 1 2 1 then we show that an ergodic mean of the quantity dx x 2 exp ith x 2 t 1 vanishes provided ge h where is well localized around the origin andge is a positivec function with support in 0 e e e f our estimate ofe f is such that the set x v v x e f may contain with probability one an infinite cluster of cubes ci which are nearest neighbours the proof is based on the technique introduced by frohlich and spencer for the analysis of the anderson model

one dimensional schrodinger operators with random or deterministic potentials new spectral types abstract we prove sufficient conditions involving only potential asymptotic near one of the infinities in order to have purely absolutely continuous components in the spectrum these deterministic results are then applied to random cases and exhibit classes of models for which with probability one one component of the spectrum is purely absolutely continuous and the rest is dense pure point with exponentially decaying eigenfunctions

power law bounds on transfer matrices and quantum dynamics in one dimension ii abstract we establish quantum dynamical lower bounds for a number of discrete one dimensional schrodinger operators these dynamical bounds are derived from power law upper bounds on the norms of transfer matrices we develop further the approach from part i and study many examples particular focus is put on models with finitely or at most countably many exceptional energies for which one can prove power law bounds on transfer matrices the models discussed in this paper include substitution models sturmian models a hierarchical model the prime model and a class of moderately sparse potentials

analyticity of density of states in a gauge invariant model for disordered electronic systems then orbital gauge invariant model of disordered electronic systems proposed by wegner is studied in the regime of dominant diagonal disorder analyticity of the density of states is established in two cases a when the number of orbitals is small b when the number of orbitals is large and the energy is in the expected extended states region

reflectionless modes in chains with large size homogeneous impurities the authors study the spectra and localization properties of tight binding chains with randomly distributed binary blocks of arbitrary size m one having site energies epsilon a and the other epsilon b respectively they demonstrate that for block size m greater than one and delta mod epsilon a epsilon b less than a critical value delta c m perfect transmission resonance modes exist in the band their number is proportional to m and they occur via dominant 1 e2 divergencies of the localization length no transmission is found for delta delta c m the results are understood by solving exactly the scattering problem from a single homogeneous impurity block of arbitrary size m in the limit of hard impurities m to infinity transmission stops only when delta delta c infinity 4v v being the intersite matrix element that is when the pure a and b bands become detached

continuous spectrum and uniform localization for ergodic schrodinger operators abstract in this paper we study localization for ergodic families of discrete schrodinger operators we prove that instability of pure point spectrum implies absence of uniform localization

bootstrap multiscale analysis and localization in random media we introduce an enhanced multiscale analysis that yields subexponentially decaying probabilities for bad events for quantum and classical waves in random media we obtain exponential decay for the resolvent of the corresponding random operators in boxes of side l with probability higher than 1 e l for any 0 1 the starting hypothesis for the enhanced multiscale analysis only requires the verification of polynomial decay of the finite volume resolvent at some sufficiently large scale with probability bigger than 1 d is the dimension note that from the same starting hypothesis we get conclusions that are valid for any 0 1 this is achieved by the repeated use of a bootstrap argument as an application we use a generalized eigenfunction expansion to obtain strong dynamical localization of any order in the hilbert schmidt norm and better estimates on the behavior of the eigenfunctions

theoremes limite pour les systemes lineaires a coefficients markoviens nous montrons un theoreme central limite et une estimee des grandes deviations pour les solutions d equations differentielles stochastiques lineaires a coefficients markoviens ces resultats sont obtenus dans le cadre general des processus markoviens multiplicatifs ils s appliquent alors par exemple au flot derive d un flot stochastique sur une variete compacte et aux produits de matrices aleatoires en dependance markovienne

fractal dimensions and the phenomenon of intermittency in quantum dynamics we exhibit an intermittency phenomenon in quantum dynamics more precisely we derive new lower bounds for the moments of order p associated to the state t e it h and averaged in time between zero and t these lower bounds are expressed in terms of generalized fractal dimensions d 1 1 p d of the measure where d is the space dimension this improves previous results obtained in terms of hausdorff and packing dimension

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