stability of the absolutely continuous spectrum of random schrodinger operators on tree graphs the subject of this work is random schrodinger operators on regular rooted tree graphs with stochastically homogeneous disorder the operators are of the form h t u v acting in l2 with t the adjacency matrix u a radially periodic potential and v a random potential this includes the only class of homogeneously random operators for which it was proven that the spectrum of h exhibits an absolutely continuous ac component a results established by a klein for weak disorder in case u 0 and v given by iid random variables on our main contribution is a new method for establishing the persistence of ac spectrum under weak disorder the method yields the continuity of the ac spectral density of h at 0 the latter is shown to converge in the l 1 sense over closed borel sets in which h 0 has no singular spectrum the analysis extends to random potentials whose values at different sites need not be independent assuming only that their joint distribution is weakly correlated across different tree branches

almost reducibility and non perturbative reducibility of quasi periodic linear systems in this paper we prove that a quasi periodic linear differential equation in sl 2 with two frequencies 1 is almost reducible provided that the coefficients are analytic and close to a constant in the case that is diophantine we get the non perturbative reducibility we also obtain the reducibility and the rotations reducibility for an arbitrary irrational under some assumption on the rotation number and give some applications for schrodinger operators our proof is a generalized kam type iteration adapted to all irrational

inverse spectral theory for random jacobi matrices we give necessary and sufficient conditions for a herglotz function to be thew function of a random stationary jacobi matrix

localization for quasiperiodic operators with unbounded monotone potentials abstract we establish non perturbative anderson localization for a wide class of 1d quasiperiodic operators with unbounded monotone potentials extending the classical results on maryland model and perturbative results for analytic potentials by bellissard lima and scoppola

almost everywhere positivity of the lyapunov exponent for the doubling map we show that discrete one dimensional schrodinger operators on the half line with ergodic potentials generated by the doubling map on the circle v n f 2 n may be realized as the half line restrictions of a non deterministic family of whole line operators as a consequence the lyapunov exponent is almost everywhere positive and the absolutely continuous spectrum is almost surely empty

analyticity of the density of states in the anderson model on the bethe lattice let h 1 2 v on l2 b whereb is the bethe lattice andv x xeb are i i d r v s with common probability distribution it is shown that for distributions sufficiently close to the cauchy distribution the density of states e is analytic in a strip about the real axis

embedding of analytic quasi periodic cocycles into analytic quasi periodic linear systems and its applications in this paper we prove that any analytic quasi periodic cocycle close to constant is the poincare map of an analytic quasi periodic linear system close to constant which bridges both methods and results in quasi periodic linear systems and cocycles we also show that the almost reducibility of an analytic quasi periodic linear system is equivalent to the almost reducibility of its corresponding poincare cocycle by the local embedding theorem and the equivalence we transfer the recent local almost reducibility results of quasi periodic linear systems hou and you in invent math 190 209 260 2012 to quasi periodic cocycles and the global reducibility results of quasi periodic cocycles avila in almost reducibility and absolute continuity 2010 avila et al in geom funct anal 21 1001 1019 2011 to quasi periodic linear systems finally we give a positive answer to a question of avila et al geom funct anal 21 1001 1019 2011 and use it to study point spectrum of long range quasi periodic operator with liouvillean frequency the embedding also holds for some nonlinear systems

factorization of random jacobi operators and backlund transformations we show that a positive definite random jacobi operatorl over an abstract dynamical systemt x x can be factorized asl d2 whered is again a random jacobi operator but defined over a new dynamical systems y y which is an integral extension oft an isospectral random toda deformation ofl corresponds to an isospectral random volterra deformation ofd the factorization leads to commuting backlund transformations which can be written explicitly in terms of titchmarsh weyl functions in the periodic case the backlund transformations are time 1 maps of a toda flow with a time dependent hamiltonian

periodic jacobi matrices on trees abstract we begin the systematic study of the spectral theory of periodic jacobi matrices on trees including a formal definition the most significant result that appears here for the first time is that these operators have no singular continuous spectrum we review important previous results of sunada and aomoto and present several illuminating examples we present many open problems and conjectures that we hope will stimulate further work

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

anderson localization for the discrete one dimensional quasi periodic schrodinger operator with potential defined by a gevrey class function in this paper we consider the discrete one dimensional schrodinger operator with quasi periodic potential vn v x n we assume that the frequency satisfies a strong diophantine condition and that the function v belongs to a gevrey class and it satisfies a transversality condition under these assumptions we prove in the perturbative regime that for large disorder and for most frequencies the operator satisfies anderson localization moreover we show that the associated lyapunov exponent is positive for all energies and that the lyapunov exponent and the integrated density of states are continuous functions with a certain modulus of continuity we also prove a partial nonperturbative result assuming that the function v belongs to some particular gevrey classes

singular components of spectral measures for ergodic jacobi matrices for ergodic 1d jacobi operators we prove that the random singular components of any spectral measure are almost surely a s mutually disjoint as long as one restricts to the set of positive lyapunov exponent in the context of extended harper s equation this yields the first rigorous proof of the thouless formula for the lyapunov exponent in the dual regions

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

exponential decay of green s functions for a class of long range hamiltonians we consider a class of long range hamiltonians with diagonal disorder onl2 z for anyergodic potentialv with non empty essential range we prove the exponential decay of the green s functions for energies in the essential range ifv is independent identically distributed we obtain the exponential decay of the green s functions for all coupling constant 0 moreover the hamiltonian has only pure point spectrum

some connections between excursion theory and the discrete schrodinger equation with random potentials we use here discrete excursion theory and the ray knight theorem in order to study questions concerning the analyticity of the density of states at low disorder as well as its smoothness properties in one dimension

localization in general one dimensional random systems i jacobi matrices we consider random discrete schrodinger operators in a strip with a potentialv n n a label in and a finite label across the strip andv an ergodic process we prove thath0 v has only point spectrum with probability one under two assumptions 1 theconditional distribution of v n n 0 1 all conditioned on v n 0 1 all has an absolutely continuous component with positive probability 2 for a e e no lyaponov exponent is zero

second order consensus in time delayed networks based on periodic edge event driven control abstract the second order consensus problems in multi agent systems with undirected topologies and time invariant delays are considered in this paper the time delay robustness of a consensus protocol based on periodic edge event driven control is investigated and the studied protocol can largely reduce the communication and controller updating costs the relationship between time delays and event detecting period is characterized by lyapunov methods it is shown that the protocol can solve state consensus problems when the interaction topology is connected and the system does not exhibit zeno behavior finally simulations are given to demonstrate the effectiveness of our theoretical results

almost periodic jacobi matrices associated with julia sets for polynomials let be the jacobi matrix associated with polynomialt z of degreen 2 the spectrum of is the julia set associated witht z which in many cases is a cantor set let 1 denote the result of omitting the first row and column ofj then it is shown that the spectrum of 1 may be purely discrete

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

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

the ten martini problem we prove the conjecture known as the ten martini problem after kac and simon that the spectrum of the almost mathieu operator is a cantor set for all nonzero values of the coupling and all irrational frequencies

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

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

absolutely continuous spectrum of multifrequency quasiperiodic schrodinger operator abstract in this paper we prove that for any d frequency analytic quasiperiodic schrodinger operator if the frequency is weak liouvillean and the potential is small enough then the corresponding operator has absolutely continuous spectrum moreover in the case d 2 we even establish the existence of ac spectrum under small potential and some super liouvillean frequency and this result is optimal due to a recent counterexample of avila and jitomirskaya 6

lyapunov exponents of the schrodinger equation with quasi periodic potential on a strip we prove that all the non negative lyapunov exponents of difference schrodinger equation y n 1 q n y n y n 1 0 infty n infty are strictly positive herey n er m andq n is a symmetricm m matrix whose offdiagonal elements do not depend onn and the diagonal elements are quasiperiodic functions q nj theta lambda f j e 2 pi i theta n alpha e with allf i non constant analytic functions sufficiently large and any irrational number

log holder continuity of the integrated density of states for stochastic jacobi matrices we consider the integrated density of states k e of a general operator on l2 v of the formh h0 v where h 0 u n sum limits left i right 1 u n i and vu n v n u n wherev is a general bounded ergodic stationary process on v we show that k e k e c log e e 1 when e e 1 2 the key is a thouless formula for the strip

perturbations of orthogonal polynomials with periodic recursion coefficients the results of denisov rakhmanov szeg shohat nevai and killip simon are extended from asymptotically constant orthogonal polynomials on the real line oprl and unit circle opuc to asymptotically periodic oprl and opuc the key tool is a characterization of the isospectral torus that is well adapted to the study of perturbations

counterexamples to the kotani last conjecture for continuum schrodinger operators via character automorphic hardy spaces abstract the kotani last conjecture states that every ergodic operator in one space dimension with non empty absolutely continuous spectrum must have almost periodic coefficients this statement makes sense in a variety of settings for example discrete schrodinger operators jacobi matrices cmv matrices and continuum schrodinger operators in the main body of this paper we show how to construct counterexamples to the kotani last conjecture for continuum schrodinger operators by adapting the approach developed by volberg and yuditskii to construct counterexamples to the kotani last conjecture for jacobi matrices this approach relates the reflectionless operators associated with the prescribed spectrum to a family of character automorphic hardy spaces and then relates the shift action on the space of operators to the resulting action on the associated characters the key to our approach is an explicit correspondence between the space of continuum reflectionless schrodinger operators associated with a given set and the space of reflectionless jacobi matrices associated with a derived set once this correspondence is established we can rely to a large extent on the previous work of volberg and yuditskii to produce the resulting action on the space of characters we analyze this action and identify situations where we can observe absolute continuity without almost periodicity in the appendix we show how to implement this strategy and obtain analogous results for extended cmv matrices

stochastic schrodinger operators and jacobi matrices on the strip we discuss stochastic schrodinger operators and jacobi matrices with wave functions taking values in l so there are 2l lyaponov exponents 1 l 0 l 1 2l 1 our results include the fact that if 1 0 on a set positive measure thenv is deterministic and one that says that e exactly 2j s are zero is the essential support of the a c spectrum of multiplicity 2j

generic singular spectrum for ergodic schrodinger operators we consider schrodinger operators with ergodic potential v n f t n n z where t is a nonperiodic homeomorphism we show that for generic f c the spectrum has no absolutely continuous component the proof is based on approximation by discontinuous potentials which can be treated via kotani theory

limit periodic schrodinger operators with a discontinuous lyapunov exponent abstract we construct a limit periodic schrodinger operator for which the lyapunov exponent has a positive measure set of discontinuities we also show that the limit periodic potentials for which the lyapunov exponent is discontinuous are dense in the space of limit periodic potentials

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

one dimensional schrodinger operators with random decaying potentials we investigate the spectrum of the following random schrodinger operators h omega frac d 2 dt 2 a t f x t omega wheref xt is a markovian potential studied by the russian school 8 we completely describe the transition of the spectrum from pure point type to absolutely continuous type as the decreasing order ofa t grows this is an extension to a continuous case of the result due to delyon simon souillard 6 who deal with the lattice case

trace class perturbations and the absence of absolutely continuous spectra we show that various hamiltonians and jacobi matrices have no absolutely continuous spectrum by showing that under a trace class perturbation they become a direct sum of finite matrices

equality of the spectral and dynamical definitions of reflection for full line jacobi matrices schrodinger operators and cmv matrices we show that being reflectionless in the sense of the well known property of m functions is equivalent to a lack of reflection in the dynamics in the sense that any state that goes entirely to x as t goes entirely to x as t this allows us to settle a conjecture of deift and simon from 1983 regarding ergodic jacobi matrices

a supersymmetric transfer matrix and differentiability of the density of states in the one dimensional anderson model let h v on 2 z where v x 9 xez are d r v s with common probability distribution v let h t e itvdv v and let k e be the integrated density of states it is proven i if h is n times differentiable with h j t 0 l t for some 0 7 0 l n then k e is a cn function in particular if v has compact support and h t 0 l h 1 with 0 then k e is c this allows v to be singular continuous ii lfh t 0 e a tl for some 0 then k e is analytic in a strip about the real axis the proof uses the supersymmetric replica trick to rewrite the averaged green s function as a two point function of a one dimensional supersymmetric field theory which is studied by the transfer matrix method

scattering of a wave packet by an interval of random medium this paper deals with the scattering theory for the one dimensional discrete schrodinger equation with a random potential having large support the main result is that a fluctuation deep within the scattering region has a very small effect on the scattering of wave packets the region of random potential is effectively opaque

isospectral deformations of random jacobi operators we show the integrability of infinite dimensional hamiltonian systems obtained by making isospectral deformations of random jacobi operators over an abstract dynamical system the time 1 map of these so called random toda flows can be expressed by aqr decomposition

analogs of the m function in the theory of orthogonal polynomials on the unit circle we show that the multitude of applications of the weyl titchmarsh m function leads to a multitude of different functions in the theory of orthogonal polynomials on the unit circle that serve as analogs of the m function

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

reducibility or nonuniform hyperbolicity for quasiperiodic schrodinger cocycles we show that for almost every frequency r q for every c o potential v r z r and for almost every energy e the corresponding quasiperiodic schr nodinger cocycle is either reducible or nonuniformly hyperbolic this result gives very good control on the absolutely continuous part of the spectrum of the corresponding quasiperiodic schr nodinger operator and allows us to complete the proof of the aubry andr le conjecture on the measure of the spectrum of the almost mathieu operator

the rotation number for finite difference operators and its properties we discuss a rotation number for second order finite difference operators ifk denotes the integrated density of states thenk 2 for almost periodic operators k is proved to lie in the frequency module whenever is outside the spectrum this yields a labelling of the gaps of the spectrum

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

absolute continuity of the integrated density of states for the almost mathieu operator with non critical coupling we show that the integrated density of states of the almost mathieu operator is absolutely continuous if and only if the coupling is non critical we deduce for subcritical coupling that the spectrum is purely absolutely continuous for almost every phase settling the measure theoretical case of problem 6 of barry simon s list of schrodinger operator problems for the twenty first century

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

support theorems for random schrodinger operators in the multi dimensional case it is shown that the increase of the topological support of the probability measure describing the randomness of potentials implies the increase of the spectrum in the one dimensional case the converse statement for the absolutely continuous spectrum is valid especially the spectrum in general dimension and the absolutely continuous spectrum in one dimension are determined only by the topological support of the random potentials

smoothness of the density of states in the anderson model at high disorder we prove smoothness of the density of states in the anderson model at high disorder for a class of potential distributions that include the uniform distribution