regularity of the invariant measure and of the density of states in the one dimensional anderson model abstract let h 1 2 v on l2 z where v x x z are i i d r v s with common probability distribution let h t e itvd v let n e be the integrated density of states and let e be the invariant measure at energy e it is proven 1 1 let h be d times differentiable on 0 with bounded derivatives such that h j t 0 as t for j 0 1 d then i n e is a cd function ii if d 3 e is absolutely continuous for all e r and d e dx is of class c d 1 2 1 jointly in e and x 2 2 let h be differentiable on 0 with h j t o e b t for j 0 1 and some b 0 then d e dx is jointly analytic in e and x

hamiltonians on discrete structures jumps of the integrated density of states and uniform convergence we study equivariant families of discrete hamiltonians on amenable geometries and their integrated density of states ids we prove that the eigenspace of a fixed energy is spanned by eigenfunctions with compact support the size of a jump of the ids is consequently given by the equivariant dimension of the subspace spanned by such eigenfunctions from this we deduce uniform convergence w r t the spectral parameter of the finite volume approximants of the ids our framework includes quasiperiodic operators on delone sets periodic and random operators on quasi transitive graphs and operators on percolation graphs

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

holder continuity of the spectral measures for one dimensional schrodinger operator in exponential regime avila and jitomirskaya prove that the spectral measure v xf of quasi periodic schrodinger operator is 1 2 holder continuous with appropriate initial vector f if satisfies diophantine condition and is small in the present paper the conclusion is extended to that for all with the spectral measure v xf is 1 2 holder continuous with small if v is real analytic in a neighbor of x c where c is a large absolute constant in particular the spectral measure xf of almost mathieu operator is 1 2 holder continuous if e c with c a large absolute constant

the uniqueness of the integrated density of states for the schrodinger operators with magnetic fields the integrated density of states ids for the schrodinger operators is defined in two ways by using the counting function of eigenvalues of the operator restricted to bounded regions with appropriate boundary conditions or by using the spectral projection of the whole space operator a sufficient condition for the coincidence of the two definitions above is given moreover a sufficient condition for the coincidence of the ids for the dirichlet boundary conditions and the ids for the neumann boundary conditions is given the proof is based only on the fundamental items in functional analysis such as the min max principle etc

regularity of the maxwell equations in heterogeneous media and lipschitz domains abstract this note establishes regularity estimates for the solution of the maxwell equations in lipschitz domains with non smooth coefficients and minimal regularity assumptions the argumentation relies on elliptic regularity estimates for the poisson problem with non smooth coefficients

harmonic analysis on sl 2 r and smoothness of the density of states in the one dimensional anderson model we consider infinite jacobi matrices with ones off diagonal and independent identically distributed random variables with distributionf v dv on diagonal iff has compact support and lies in some sobolev spacel 1 then we prove that the integrated density of states k e isc ine

some remarks on discrete aperiodic schrodinger operators we consider schrodinger operators onl2 with deterministic aperiodic potential and schrodinger operators on the l2 space of the set of vertices of penrose tilings and other aperiodic self similar tilings the operators onl2 fit into the formalism of ergodic random schrodinger operators hence their lyapunov exponent integrated density of states and spectrum are almost surely constant we show that they are actually constant the lyapunov exponent for one dimensional schrodinger operators with potential defined by a primitive substitution the integrated density of states and the spectrum in arbitrary dimension if the system is strictly ergodic we give examples of strictly ergodic schrodinger operators that include several kinds of almost periodic operators that have been studied in the literature for schrodinger operators on penrose tilings we prove that the integrated density of states exists and is independent of boundary conditions and the particular penrose tiling under consideration

homogenization of the maxwell equations at fixed frequency the homogenization of the maxwell equations at fixed frequency is addressed in this paper the bulk homogenized electric and magnetic properties of a material with a periodic microstructure are found from the solution of a local problem on the unit cell by suitable averages the material can be anisotropic and satisfies a coercivity condition the exciting field is generated by an incident field from sources outside the material under investigation a suitable sesquilinear form is defined for the interior problem and the exterior calderon operator is used to solve the exterior radiating fields the concept of two scale convergence is employed to solve the homogenization problem a new a priori estimate is proved as well as a new result on the correctors

necessary and sufficient conditions for the stability of a class of 2 d recursive digital filters in this paper we propose an alternative approach for testing the stability of two dimensional causal recursive digital filters which are devoid of nonessential singularities of the second kind we make a detailed comparison of the computational efficiency of this method with two other algebraic methods available in the literature we also derive using the approach presented in this paper a simple set of necessary and sufficient conditions for the stability of a class of 2 d recursive filters

on non linear systems and algebraic geometry the classical root locus and inverse laplace transformation theories are generalised to homogeneous second order non linear systems and a connection with algebraic geometry is established

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

a short and versatile finite element multiscale code for homogenization problems we describe a multiscale finite element fe solver for elliptic or parabolic problems with highly oscillating coefficients based on recent developments of the so called heterogeneous multiscale method hmm the algorithm relies on coupled macro and microsolvers the framework of the hmm allows to design a code whose structure follows the classical finite elements implementation at the macro level to account for the fine scales of the problem elementwise numerical integration is replaced by micro fe methods on sampling domains we discuss a short and flexible fe implementation of the multiscale algorithm which can accommodate simplicial or quadrilateral fe and various coupling conditions for the constrained micro simulations extensive numerical examples including three dimensional and time dependent problems are presented illustrating the efficiency and the versatility of the computational strategy

on the approximation of electromagnetic fields by edge finite elements part 1 we propose sharp results on the numerical approximation of low regularity electromagnetic fields by edge finite elements we consider general geometrical settings including topologically non trivial domains or domains with a non connected boundary in the model the electric permittivity and magnetic permeability are symmetric tensor valued piecewise smooth coefficients in all cases the error can be bounded by h times a constant where h is the meshsize for some exponent 0 1 that depends both on the geometry and on the coefficients it relies either on classical estimates when 1 2 or on a new combined interpolation operator when 1 2 the optimality of the value of is discussed with respect to abstract shift theorems in some simple configurations typically for scalar valued permittivity and permeability the value of can be further characterized this paper is the first one in a series dealing with the approximation of electromagnetic fields by edge finite elements

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

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

holder continuity of absolutely continuous spectral measure for multi frequency schrodinger operators abstract we establish sharp results on the modulus of continuity of the distribution of the spectral measure for multi frequency schrodinger operators with diphantine frequencies and small analytic potentials

on the approximation of electromagnetic fields by edge finite elements part 2 in the second part of this series of papers we consider highly oscillatory media in this situation the need for a triangulation that resolves all microscopic details of the medium makes standard edge finite elements impractical because of the resulting tremendous computational load on the other hand undersampling by using a coarse mesh might lead to inaccurate results to overcome these difficulties and to improve the ratio between accuracy and computational costs homogenization techniques can be used in this paper we recall analytical homogenization results and propose a novel numerical homogenization scheme for maxwells equations in frequency domain this scheme follows the design principles of heterogeneous multiscale methods we prove convergence to the effective solution of the multiscale maxwells equations in a periodic setting and give numerical experiments in accordance to the stated results

positivity and nonnegativity conditions of a quartic equation and related problems in this paper explicit conditions for positivity no real roots nonnegativity on positive real axis no positive real roots with odd multiplicity and stability aperiodicity all roots are real and negative and simple of a quartic or biquadratic equation are given the derived conditions from the known solution of the quartic equation are not only complete but simpler than those derived from sturm extended hurwitz inners and hankel methods because of abel s theorem no explicit solution in terms of the roots of an equation higher than quartic exists similar simplification for higher degree polynomial equations may not be possible furthermore explicit conditions for positivity and nonnegativity of equations of higher degree than four are extremely difficult to obtain and may not be possible the results of the paper will hopefully shed some light on a century old problem and thus enhance the engineering application of the derived condition to higher order systems

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

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

large deviations and lifshitz singularity of the integrated density of states of random hamiltonians we consider the integrated density of states ids of random hamiltonian h v v being a random field on d which satisfies a mixing condition we prove that the probability of large fluctuations of the finite volume ids 1 h d around the thermodynamic limit is bounded from above by exp k k 0 in this case can be recovered from a variational principle furthermore we show the existence of a lifshitztype of singularity of as 0 in the case where v is non negative more precisely we prove the following bound exp k d 2 as 0 k 0 this last result is then discussed in some examples

absence of localisation in the almost mathieu equation the author considers the discrete schrodinger operator on z with the potential lambda cos 2 pi alpha n theta this one dimensional model occurs in the study of an electron in a two dimensional periodic potential with a uniform magnetic field first it is proved that for every alpha and for lambda 2 this operator has no eigenvalue i e localised state furthermore at lambda 2 the eigenvalues if they exist belong to the set where the lyapunov exponent vanishes and the associated eigenvectors at in l2 z but are not summable

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

sharp holder continuity of the lyapunov exponent of finitely differentiable quasi periodic cocycles we show that if the base frequency is diophantine then the lyapunov exponent of a c k quasi periodic sl 2 mathbb r cocycle is 1 2 holder continuous in the almost reducible regime as a consequence we show that if the frequency is diophantine and the potential is small then the integrated density of states of the corresponding quasi periodic schrodinger operator is 1 2 holder continuous

a set of necessary and sufficient stability conditions for low order two dimensional polynomials in this note a set of necessary and sufficient conditions for two dimensional polynomials which are quadratic in both variables or quartic in one and linear in the other are given in both cases the important stability condition reduces to checking the nonexistence of positive real roots of a quartic equation conditions for the latter have been recently presented by the authors 1 furthermore by appealing to coordinate transformation the explicit stability conditions are extended to the largest possible class of two dimensional polynomials finally a sufficient condition for stability is given for any two dimensional polynomial

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

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

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

random schrodinger operators arising from lattice gauge fields i existence and examples we consider new models of ergodic schrodinger operators whose existence relies on a cohomological theorem of feldman and moore in ergodic theory these operators generalize the harper operator which describes the case of a constant magnetic field an example is the case when the magnetic field is given by independent random variables attached to the lattice plaquettes a generalization of the feldman moore theorem by lind to non abelian groups also allows us to consider schrodinger operators obtained from non abelian lattice gauge fields the existence result extends to more general graphs like to operators on tilings and to higher dimensions we compute some moment expansions for the density of states for example for independent identically and uniformly distributed magnetic fields a model which has been studied at least since 1970 and whose existence can also be seen without involving the above mentioned existence theorem we show that the nth moment is the number of closed paths in the two dimensional lattice starting at the origin for which the winding number vanishes at each plaquette point this goes beyond the brinkman rice self retracing path approximation other examples are a higher dimensional example a one dimensional anderson model which can be treated in this framework as well as the hofstadter model with constant magnetic field where one averages over all possible magnetic fields we also reprove a result of jitomirskaya mandelshtam stating that the deterministic aharonov bohm model is a compact perturbation of the free laplacian

cantor spectrum for the almost mathieu equation for a dense g of pairs in r2 we prove that the operator hu n u n 1 u n 1 cos 2 n u n has a nowhere dense spectrum along the way we prove several interesting results about the case pq of which we mention a if q is not an integral multiple of then all gaps are open and b if q is even and is chosen suitably then the middle gap is closed for all

multiscale computational method for heat conduction problems of composite structures with diverse periodic configurations in different subdomains abstract this study develops a novel multiscale computational method for heat conduction problems of composite structures with diverse periodic configurations in different subdomains firstly the second order two scale sots solutions for these multiscale problems are successfully obtained based on multiscale asymptotic analysis then the error analysis of sots solutions in the pointwise sense is given to illustrate the importance of developing the sots solutions furthermore the error estimate for the sots approximate solutions in the integral sense is presented in addition a sots numerical algorithm is proposed to effectively solve these problems based on finite element method fem finally some numerical examples verify the feasibility and effectiveness of the sots numerical algorithm we proposed in this paper a unified two scale computational framework is established for heat conduction problems of composite structures with diverse periodic configurations in different subdomains

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

orthogonal polynomials of equilibrium measures supported on cantor sets the equilibrium measure of a compact set is a fundamental object in logarithmic potential theory we compute numerically this measure and its orthogonal polynomials when the compact set is a cantor set defined by an iterated function system we first construct sequences of discrete measures via the solution of large systems of non linear equations that converge weakly to the equilibrium measure successively we compute their jacobi matrices via standard procedures enhanced for the scope numerical estimates of the convergence rate to the limit jacobi matrix are provided which show stability and efficiency of the whole procedure as a companion result we also compute jacobi matrices in two other cases equilibrium measures on finite sets of intervals and balanced measures of iterated function systems these algorithms can reach large polynomial orders therefore we study the asymptotic behavior of the orthogonal polynomials and by a natural extension of the concept of regular root asymptotics we derive an efficient scheme for the computation of complex green s functions and of related conformal mappings

exact wave functions of an electron on a quasiperiodic lattice definition of an infinite dimensional riemann theta function a scheme for obtaining the exact wave functions of an electron on a quasiperiodic lattice is presented it is shown that the trace map plays a very important role for construction of the infinite dimensional riemann theta function in terms of which the wave functions can be represented

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

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

a new heterogeneous multiscale method for time harmonic maxwell s equations in this paper we suggest a new heterogeneous multiscale method hmm for the time harmonic maxwell equations in locally periodic media the method is constructed by using a divergence regularization in one of the cell problems this allows us to introduce fine scale correctors that are not subject to a cumbersome divergence free constraint and which can hence easily be implemented to analyze the method we first revisit classical homogenization theory for time harmonic maxwell equations and derive a new homogenization result that makes use of the divergence regularization in the two scale homogenized equation we then show that the hmm is equivalent to a discretization of this equation in particular writing both problems in a fully coupled two scale formulation is the crucial starting point for a corresponding numerical analysis of the method with this approach we are able to prove rigorous a priori error estimates in the mathbf h curl and the h 1 norm and we derive reliable and efficie

arithmetic spectral transitions for the maryland model we give a precise description of spectra of the maryland model h u n un 1 un 1 tan n un for all values of parameters we introduce an arithmetically defined index and show that for sc h e e and pp h e e since ac h this gives a complete description of the spectral decomposition for all values of parameters and making it the first case of a family where arithmetic spectral transition is described without any parameter exclusion the set of eigenvalues can be explicitly identified for all parameters using the quantization condition we also establish for the first time for this or any other model a quantization condition for singular continuous spectrum an arithmetically defined measure zero set that supports singular continuous measures for all parameters 2017 wiley periodicals inc

numerical treatment of realistic boundary conditions for the eddy current problem in an electrode via lagrange multipliers this paper deals with the finite element solution of the eddy current problem in a bounded conducting domain crossed by an electric current and subject to boundary conditions appropriate from a physical point of view two different cases are considered depending on the boundary data input current density flux or input current intensities the analysis of the former is an intermediate step for the latter which is more realistic in actual applications weak formulations in terms of the magnetic field are studied the boundary conditions being imposed by means of appropriate lagrange multipliers the resulting mixed formulations are analyzed depending on the regularity of the boundary data finite element methods are introduced in each case and error estimates are proved finally some numerical results to assess the effectiveness of the methods are reported

anderson localization for the 1 d discrete schrodinger operator with two frequency potential we prove the complete exponential localization of eigenfunctions for the 1 d discrete schrodinger operators with quasi periodic potentials having two basic frequencies it is shown also that for such operators there is no forbidden zones in the spectrum unlike the operators with one basic frequency

realization theory and the infinite dimensional root locus the theory of realization of linear input output maps is applied to the study of root loci of distributed systems with bounded defining operators

universality of general beta ensembles we prove the universality of the ensembles with convex analytic potentials and for any 0 i e we show that the spacing distributions of log gases at any inverse temperature coincide with those of the gaussian ensembles

computing the spectral function for singular sturm liouville problems algorithms for computing sturm liouville spectral density functions are developed based on several mathematical characterizations convergence and error bounds are derived and methods are tested on several examples the results are compared with those from the existing software package sledge

time harmonic wave diffraction problems in materials with sign shifting coefficients some electromagnetic materials present in a given frequency range an effective dielectric permittivity and or magnetic permeability which are negative we are interested in the reunion of such a negative material and a classical one more precisely we consider here a scalar model problem for the simulation of a wave transmission between two such materials this model is governed by a helmholtz equation with a weight function in the d principal part which takes positive and negative real values introducing additional unknowns we have already proposed in bonnet ben dhia et al 2006 1 some new variational formulations of this problem which are of fredholm type provided the absolute value of the contrast of permittivities is large enough and therefore suitable for a finite element discretization we prove here that under similar conditions on the contrast the natural variational formulation of the problem although not coercive plus compact is nonetheless suitable for a finite element discretization this leads to a numerical approach which is straightforward less costly than the previous ones and very accurate

absolutely continuous spectra of quasiperiodic schrodinger operators several aspects of the general and constructive spectral theory of quasiperiodic schrodinger operators in one dimension are discussed an explicit formula for the absolutely continuous a c spectral densities that yields an immediate proof of the fact that the kolmogorov arnold moser kam spectrum constructed by dinaburg sinai and russmann funkt anal prilozen 9 8 1975 ann acad sci 357 90 1980 is a subset of the a c spectrum is provided some quasiperiodicity properties of the deift simon a c eigenfunctions are proved namely that the normalized phase of such eigenfunctions is a quasiperiodic distribution in the constructive part the dinaburg sinai russmann theory is extended to quasiperiodic perturbations of periodic schrodinger operators using a kam hamiltonian formalism based on a new treatment of perturbations of harmonic oscillators particular attention is devoted to the dependence upon the eigenvalue parameter and a complete control of kam objects is achieved using the notion

holder continuity of the integrated density of states for quasi periodic schrodinger equations and averages of shifts of subharmonic functions in this paper we consider various regularity results for discrete quasiperiodic schr6dinger equations n l pn 1 v 9 nw on eon with analytic potential v we prove that on intervals of positivity for the lyapunov exponent the integrated density of states is holder continuous in the energy provided w has a typical continued fraction expansion the proof is based on certain sharp large deviation theorems for the norms of the monodromy matrices and the avalanche principle the latter refers to a mechanism that allows us to write the norm of a monodromy matrix as the product of the norms of many short blocks in the multi frequency case the integrated density of states is shown to have a modulus of continuity of the form exp log tl for some 0 a 1 but currently we do not obtain holder continuity in the case of more than one frequency we also present a mechanism for proving the positivity of the lyapunov exponent for large disorders for a general class of equations the only requirement for this approach is some weak form of a large deviation theorem for the lyapunov exponents in particular we obtain an independent proof of the herman sorets spencer theorem in the multi frequency case the approach in this paper is related to the recent nonperturbative proof of anderson localization in the quasi periodic case by j bourgain and m goldstein

global theory of one frequency schrodinger operators we study schrodinger operators with a one frequency analytic potential focusing on the transition between the two distinct local regimes characteristic respectively of large and small potentials from the dynamical point of view the transition signals the emergence of non uniform hyperbolicity so the dependence of the lyapunov exponent with respect to parameters plays a central role in the analysis though often ill behaved by conventional measures we show that the lyapunov exponent is in fact remarkably regular in a stratified sense which we define the irregularity comes from the matching of nice analytic or smooth functions along sets with complicated geometry this result allows us to establish that the critical set for the transition lies within countably many codimension one subvarieties of the infinite dimensional parameter space a more refined renormalization based analysis shows that the critical set is rather thin within those subvarieties and allows us to conclude that a typical potential has no critical energies such acritical potentials also form an open set and have several interesting properties only finitely many phase transitions may happen but never at any specific point in the spectrum and the lyapunov exponent is minorated in the region of the spectrum where it is positive on the other hand we do show that the number of phase transitions can be arbitrarily large

almost periodic potentials in higher dimensions this work was motivated by the paper of r johnson and j moser see j m in the references on the one dimensional almost periodic potentials here we study the operator l 2 q where q is an almost periodic function in r d it is shown that some of the results of j m extend to the multidimensional case our approach includes the one dimensional case as well we start with the kernel k t x y of the semigroup e tl for fixed t 0 and u r d it is known we review the proof that k t x x u is almost periodic in x with frequency module not bigger than the one of q

the density of states in the anderson model at weak disorder a renormalization group analysis of the hierarchical model we study the density of states in a hierarchical approximation of the anderson tight binding model at weak disorder using a renormalization group approach since the laplacian term in our model is hierarchical the renormalization group transformations act essentially on the local potential distribution and the energy technically we use the supersymmetric replica trick and study the averaged green s function starting with a gaussian distribution with small variance we find that the density of states is analytic as soon as the variance of the potential is turned on except possibly near the band edge where we can show this only for 2 which corresponds tod 4 moreover it is perturbatively close to the free one except near the eigenvalues of the hierarchical laplacian where it is given up to perturbative corrections by the rescaled potential distribution

algebro geometric solutions of the baxter szegő difference equation we derive theta function representations of algebro geometric solutions of a discrete system governed by a transfer matrix associated with an extension of the trigonometric moment problem studied by szeg and baxter we also derive a new hierarchy of coupled nonlinear difference equations satisfied by these algebro geometric solutions

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

finite element methods with matching and nonmatching meshes for maxwell equations with discontinuous coefficients we investigate the finite element methods for solving time dependent maxwell equations with discontinuous coefficients in general three dimensional lipschitz polyhedral domains both matching and nonmatching finite element meshes on the interfaces are considered and optimal error estimates for both cases are obtained the analysis of the latter case is based on an abstract framework for nested saddle point problems along with a characterization of the trace space for h curl d a new extension theorem for h curl d functions in any lipschitz domain d and a novel compactness argument for deriving discrete inf sup conditions

anderson transitions for a family of almost periodic schrodinger equations in the adiabatic case this work is devoted to the study of a family of almost periodic one dimensional schrodinger equations using results on the asymptotic behavior of a corresponding monodromy matrix in the adiabatic limit we prove the existence of an asymptotically sharp anderson transition in the low energy region more explicitly we prove the existence of energy intervals containing only singular spectrum and of other energy intervals containing absolutely continuous spectrum the zones containing singular spectrum and those containing absolutely continuous are separated by asymptotically sharp transitions the analysis may be viewed as utilizing a complex wkb method for adiabatic perturbations of periodic schrodinger equations the transition energies are interpreted in terms of phase space tunneling

spectral properties of a tight binding hamiltonian with period doubling potential we study a one dimensional tight binding hamiltonian with a potential given by the period doubling sequence we prove that its spectrum is purely singular continuous and supported on a cantor set of zero lebesgue measure for all nonzero values of the potential strength moreover we obtain the exact labelling of all spectral gaps and compute their widths asymptotically for small potential strength

schrodinger operators in the twentieth century this paper reviews the past fifty years of work on spectral theory and related issues in nonrelativistic quantum mechanics

lifschitz tails for the anderson model we present in an expository way an elementary rigorous proof patterned after an argument of kirsch martinelli that the anderson model has lifschitz tails in very great generality

the local microscale problem in the multiscale modeling of strongly heterogeneous media effects of boundary conditions and cell size many multiscale methods are based on the idea of extracting macroscopic behavior of solutions by solving an array of microscale models over small domains a key ingredient in such multiscale methods is the boundary condition and the size of the computational domain over which the microscale problems are solved this problem is systematically investigated in the present paper in the context of modeling strongly heterogeneous media three different boundary conditions are considered the periodic boundary condition dirichlet boundary condition and the neumann boundary condition each is applied to several benchmark problems the random checker board problem periodic problem with isotropic macroscale behavior periodic problem with anisotropic macroscale behavior and periodic laminated media in each case convergence studies are conducted as the domain size for the microscale problem is changed convergence rates as well as the size of fluctuations in the computed effective coefficients are compared for the different formulations in addition we will discuss a mixed dirichlet neumann boundary condition that is often used in porous medium modeling we explain why that leads to unsatisfactory results and how it can be corrected also discussed are the different averaging methods used in extracting the effective coefficients

continuity of the lyapunov exponent for quasiperiodic operators with analytic potential we study regularity properties of the lyapunov exponent l of one frequency quasiperiodic operators with analytic potential under no assumptions on the diophantine class of the frequency we prove joint continuity of l in frequency and energy at every irrational frequency

on the measure of the spectrum for the almost mathieu operator we obtain partial results on the conjecture that for the almost mathieu operator at irrational frequency the measure of the spectrum s 4 2 for 2 we show that if n is rational and alpha n to alpha irrational then s alpha n lambda theta to 4 2 lambda

a difference equation arising from the trigonometric moment problem having random reflection coefficients an operator theoretic approach abstract operators associated with polynomials orthogonal on the unit circle and their doubly infinite analogs are considered we assume that the elements in the operators are obtained from a stationary stochastic process and we investigate the properties of the spectrum of these operators lyapunov exponents are introduced and we examine their relationship with the absolutely continuous components of the spectrum of the above operators

orthogonal polynomials on the unit circle via a polynomial mapping on the real line let f n n 0 be a sequence of monic orthogonal polynomials on the unit circle opuc with respect to a symmetric and finite positive borel measure email protected on 0 email protected and let 1 a 0 a 1 a 2 be the associated sequence of verblunsky coefficients in this paper we study the sequence email protected n n 0 of monic opuc whose sequence of verblunsky coefficients is 1 b 1 b 2 b n 1 a 0 b n 1 b 2 b 1 a 1 b 1 b 2 b n 1 a 2 b n 1 b 2 b 1 a 3 where b 1 b 2 b n 1 are n 1 fixed real numbers such that b email protected 1 1 for all j 1 2 n 1 so that email protected n n 0 is also orthogonal with respect to a symmetric and finite positive borel measure email protected on the unit circle we show that the sequences of monic orthogonal polynomials on the real line oprl corresponding to f n n 0 and email protected n n 0 by szego s transformation are related by some polynomial mapping giving rise to a one to one correspondence between the monic opuc email protected n n 0 on the unit circle and a pair of monic oprl on a subset of the interval 1 1 in particular we prove email protected q email protected n 1 q email protected q n q email protected q n q q n q supported on a subset of the union of 2n intervals contained in 0 email protected such that any two of these intervals have at most one common point and where up to an affine change in the variable z n 1 and email protected n are algebraic polynomials in email protected of degrees n 1 and n respectively defined only in terms of a 0 b 1 b n 1 this measure induces a measure on the unit circle supported on the union of 2n arcs pairwise symmetric with respect to the real axis the restriction to symmetric measures or real verblunsky coefficients is needed in order that szego s transformation may be applicable

high dimensional finite elements for two scale maxwell wave equations abstract we develop an essentially optimal numerical method for solving two scale maxwell wave equations in a domain d r d the problems depend on two scales one macroscopic scale and one microscopic scale solving the macroscopic two scale homogenized problem we obtain the desired macroscopic and microscopic information this problem depends on two variables in r d one for each scale that the original two scale equation depends on and is thus posed in a high dimensional tensorized domain the straightforward full tensor product finite element fe method is exceedingly expensive we develop the sparse tensor product fes that solve this two scale homogenized problem with essentially optimal number of degrees of freedom i e the number of degrees of freedom differs by only a logarithmic multiplying factor from that required for solving a macroscopic problem in a domain in r d only for obtaining a required level of accuracy numerical correctors are constructed from the fe solution we derive a rate of convergence for the numerical corrector in terms of the microscopic scale and the fe mesh width numerical examples confirm our analysis

inverse problem for polynomials orthogonal on the unit circle polynomials orthogonal on the unit circle with random recurrence coefficients and finite band spectrum are investigated it is shown that the coefficients are in fact quasi periodic the measures associated with these quasi periodic coefficients are exhibited and necessary and sufficient conditions relating quasi periodicity and spectral measures of this type are given analogs for polynomials orthogonal on subsets of the real line are also presented

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