a bias correction method for identification of linear dynamic errors in variables models this paper considers the problem of identifying linear systems where the input is observed in white noise but the output is observed in colored noise which also includes process disturbances an efficient method is developed which can perform unbiased parameter estimation without utilizing a prefilter the developed method is characterized by attractive features direct use of the observed data without prefiltering no need to evaluate autocorrelation functions for the input noise no need to identify a high order augmented system and provision of a direct unbiased estimate of the system parameters without parameter extraction computer simulations are presented to illustrate its superior performance including its significantly reduced computational complexity

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

brief paper accuracy analysis of bias eliminating least squares estimates for errors in variables systems the bias eliminating least squares bels method is one of the consistent estimators for identifying dynamic errors in variables systems in this paper we investigate the accuracy properties of the bels estimates an explicit expression for the normalized asymptotic covariance matrix of the estimated parameters is derived and supported by some numerical examples

errors in variables identification using maximum likelihood estimation in the frequency domain this paper deals with the identification of errors in variables eiv models corrupted by additive and uncorrelated white gaussian noises when the noise free input is an arbitrary signal not required to be periodic in particular a frequency domain maximum likelihood ml estimator is proposed and analyzed in some detail as some other eiv estimators this method assumes that the ratio of the noise variances is known the estimation problem is formulated in the frequency domain it is shown that the parameter estimates are consistent an explicit algorithm for computing the asymptotic covariance matrix of the parameter estimates is derived the possibility to effectively use lowpass filtered data by using only part of the frequency domain is discussed analyzed and illustrated

the frisch scheme in dynamic system identification abstract the use of the frisch scheme in the identification of linear dynamic systems is investigated in order to describe the whole family of models that can explain given input output noisy sequences unlike the algebraic case it is shown that in general only a single model is compatible with the data these results are first proposed for single input single output systems and then generalized to the multivariable case

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

accuracy analysis of the frisch scheme for identifying errors in variables systems several estimation methods have been proposed for identifying errors in variables systems where both input and output measurements are corrupted by noise one of the promising approaches is the so called frisch scheme this paper provides an accuracy analysis of the frisch scheme applied to system identification the estimates of the system parameters and the noise variances are shown to be asymptotically gaussian distributed an explicit expression for the covariance matrix of the asymptotic distribution is given as well numerical simulations support the theoretical results a comparison with the cramer rao lower bound is also given in the examples and it is shown that the frisch scheme gives a performance close to the cramer rao bound for large signal to noise ratios snrs

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

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

1d quasiperiodic operators latent symmetries a large class of discrete quasiperiodic operators is shown to be decomposed into orbits ofsl 2 z action with equal densities of states moreover under some natural assumptions all nontrivial representatives of the mentioned action transform operators with pure point spectrum into those with absolutely continuous spectrum some applications of these results are presented

a generalized instrumental variable estimation method for errors in variables identification problems the errors in variables identification problem concerns dynamic systems whose input and output variables are affected by additive noise several estimation methods have been proposed for identifying dynamic errors in variables models in this paper it is shown how a number of common methods for errors in variables methods can be put into a general framework resulting into a generalized instrumental variable estimator give various computational aspects of give are presented and the asymptotic distribution of the parameter estimates is derived

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

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

identification of eiv models by compensated pem this paper deals with identification of discrete time errors in variables models where the input and output data are both perturbed by different additive noises the goal is to study the effects of

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

a generalised instrumental variable estimator for multivariable errors in variables identification problems a new and very general estimation method the generalised instrumental variable estimator give has been introduced to deal with the errors in variables identification the give contains several well known estimation methods as special cases that can be obtained by various user choices in give these methods include the bias eliminating least squares method a number of different versions of the frisch scheme and the extended compensated least squares method this article shows how previous analysis for single input single output systems can be extended in a non trivial way to also cope with multi input multi output systems various computational aspects of give are presented and it is shown how the parameter estimates can be computed in an efficient way the asymptotic distribution of the parameter estimates is also derived and analysed

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

novel parameter estimation of autoregressive signals in the presence of noise this paper proposes a new method for estimating the parameters of an autoregressive ar signal from observations corrupted by white noise the feature of the new method is that the observation noise variance estimate is converted into the only solution of a nonlinear equation to yield unbiased estimate of the ar parameters moreover a convergent newton iterative algorithm with a deterministic initial point is presented for efficient implementation of the proposed new estimation method as a result the proposed new method can minimize the error of estimating the variance of the observation noise since more accurate estimates of this observation noise variance can be attained at earlier stages the proposed method can achieve a good performance in estimating the ar signal parameters numerical results demonstrate that the proposed new algorithm is more effective in terms of accuracy and robustness against noise than conventional algorithms

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

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

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

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

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

extended accuracy analysis of a covariance matching approach for identifying errors in variables systems a system identification method for errors in variables problems based on covariance matching was recently proposed in the first step a small amount of covariances of noisy input output data are computed and then a parametric model is fitted to these covariances in this paper the method is further analyzed and the asymptotic accuracy of the parameter estimates is derived an explicit algorithm for computing the asymptotic covariance matrix of the parameter estimates is given and the identification method is shown to be asymptotically statistically efficient assuming that the given information is the computed covariances as an important byproduct an efficient algorithm is presented for computing the covariance matrix of the computed input output covariances

rotation number associated with difference equations satisfied by polynomials orthogonal on the unit circle abstract the techniques of topological dynamics and differential dynamical systems are used to study polynomials orthogonal with respect to a measure supported on the unit circle it is assumed that the reflection coefficients associated with these polynomials form a stationary stochastic ergodic process in particular the techniques mentioned above are used to prove a gap labelling result

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

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

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

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

a covariance matching approach for identifying errors in variables systems the errors in variables identification problem concerns dynamic systems whose input and output variables are affected by additive noise several estimation methods have been proposed for identifying dynamic errors in variables models in this paper a covariance matching approach is proposed to solve the identification problem it applies for general types of input signals the method utilizes a small set of covariances of the measured input output data this property applies also for some other methods such as the frisch scheme and the bias eliminating least squares method algorithmic details for the proposed method are provided user choices for example specification of which input output covariances to utilize are discussed in some detail the method is evaluated by using numerical examples and is shown to have competitive properties as compared to alternative methods

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

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

brief paper relations between bias eliminating least squares the frisch scheme and extended compensated least squares methods for identifying errors in variables systems there are many methods for identifying errors in variables systems among them bias eliminating least squares bels the frisch scheme and extended compensated least squares ecls methods are attractive approaches because of their simplicity and good estimation accuracy these three methods are all based on a bias compensated least squares bcls principle in this paper the relationships between them are considered in particular the set of nonlinear equations utilized in these three methods are proved to be equivalent under different noise conditions also for finite samples it is shown that bels frisch and ecls methods have the same asymptotic estimation accuracy providing the same extended vector is used

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

comparing some classes of bias compensating least squares methods three different classes of bias compensating least squares identification methods are compared and shown to be identical it is also discussed how the user parameters in the classes can be chosen to achieve optimal accuracy of the parameter estimates

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

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

new consistent methods for order and coefficient estimation of continuous time errors in variables fractional models the errors in variables identification problem concerns dynamic systems in which input and output signals are contaminated by an additive noise several estimation methods have been proposed for identifying dynamic errors in variables rational models this paper presents new consistent methods for order and coefficient estimation of continuous time systems by errors in variables fractional models first differentiation orders are assumed to be known and only differential equation coefficients are estimated two estimators based on higher order statistics third order cumulants are developed the fractional third order based least squares algorithm ftocls and the fractional third order based iterative least squares algorithm ftocils then they are extended using a nonlinear optimization algorithm to estimate both the differential equation coefficients and the commensurate order the performances of the proposed algorithms are illustrated with a numerical example

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

finite groups whose non linear irreducible characters of the same degree are galois conjugate abstract we classify the finite groups whose non linear irreducible characters that are not conjugate under the natural galois action have distinct degrees therefore extending the results in berkovich et al 1 and dolfi et al 4

discrete magnetic laplacian we consider a 2 dimensional discrete operator which we call the discrete magnetic laplacian dml it is an analogue of the magnetic schrodinger operator it follows from well known arguments that dml has the same spectrum as a subset inr as the almost mathieu operator am they also have the same integrated density of states ids which is known to be continuous we show that dml is an element in a ii1 factor and its ids can be expressed through the trace in the ii1 factor it follows that dml never has anyl2 eigenfunctions i e has no point spectrum then we formulate a natural algebraic conjecture which implies that the spectrum of dml hence the spectrum of am is a cantor set

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

on the integrated density of states for schrodinger operators on ℤ2 with quasi periodic potential in this paper we consider discrete schrodinger operators on the lattice 2 with quasi periodic potential we establish new regularity results for the integrated density of states as well as a quantitative version of a thouless formula as previously considered by craig and simon for real energies and with rates of convergence the main ingredient is a large deviation theorem for the green s function that was recently established by bourgain goldstein and the author for the integrated density of states an argument of bourgain is used finally we establish certain fine properties of separately subharmonic functions of two variables that might be of independent interest

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

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

a unified framework for eiv identification methods when the measurement noises are mutually correlated in this paper the previously introduced generalized instrumental variable estimator give is extended to the case of errors in variables models where the additive input and output noises are mutually correlated white processes it is shown how many estimators proposed in the literature can be described as various special cases of a generalized instrumental variable framework it is also investigated how to analyze the common situation where some of the equations that define the estimator are to hold exactly and others to hold approximately in a least squares sense providing a detailed study of the accuracy 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

frequency domain identification of fir models in the presence of additive input output noise this paper describes a new approach for identifying fir models from a finite number of measurements in the presence of additive and uncorrelated white noise in particular two different frequency