the absolutely continuous spectrum of jacobi matrices i explore some consequences of a groundbreaking result of brei messer and pearson on the absolutely continuous spectrum of one dimensional schrodinger operators these include an oracle theorem that predicts the potential and rather general results on the approach to certain limit potentials in particular we prove a denisov rakhmanov type theorem for the general nite gap case the main theme is the following it is extremely dicult to produce abso lutely continuous spectrum in one space dimension and thus its existence has strong implications

stability of spectral types for jacobi matrices under decaying random perturbations abstract we study stability of spectral types for semi infinite self adjoint tridiagonal matrices under random decaying perturbations we show that absolutely continuous spectrum associated with bounded eigenfunctions is stable under hilbert schmidt random perturbations we also obtain some results for singular spectral types

eigenfunctions transfer matrices and absolutely continuous spectrum of one dimensional schrodinger operators on l r and its half line problem h on l2 0 with u 0 0 boundary conditions we will focus on a new approach to the absolutely continuous spectrum ac h and more generally ac h the essential support of the a c part of the spectral measures what is new in our approach is that it relies on estimates on the transfer matrix that is the 2 2 matrix te n m which takes u m 1 u m to u n 1 u n for solutions u of hu eu in the continuum case use u x u x instead of u x 1 u x we let te n te n 0 for example we will prove the following

schrodinger operators with sparse potentials asymptotics of the fourier transform of the spectral measure we study the pointwise behavior of the fourier transform of the spectral measure for discrete one dimensional schrodinger operators with sparse potentials we find a resonance structure which admits a physical interpretation in terms of a simple quasiclassical model we also present an improved version of known results on the spectrum of such operators

a probabilistic approach to one dimensional schrodinger operators with sparse potentials we consider the one dimensional schrodinger equation with sparse potential v i e mainly v 0 it is shown that the asymptotics of the solutions corresponding to positive energies e can be approximately described by an infinite sum of independent random variables using results from probability theory we can then determine the spectral properties of the operators under consideration we prove absolute continuity for a general class of potentials and we also have examples with singular continuous spectrum

absolutely continuous spectrum of dirac operators with square integrable potentials we show that the absolutely continuous part of the spectral function of the one dimensional dirac operator on a half line with a constant mass term and a real square integrable potential is strictly increasing throughout the essential spectrum 1 1 the proof is based on estimates for the transmission coefficient for the full line scattering problem with a truncated potential and a subsequent limiting procedure for the spectral function furthermore we show that the absolutely continuous spectrum persists when an angular momentum term is added thus also establishing the result for spherically symmetric dirac operators in higher dimensions

asymptotics of the fourier transform of the spectral measure for schrodinger operators with bounded and unbounded sparse potentials we study the pointwise behaviour of the fourier transform of the spectral measure for discrete one dimensional schrodinger operators with unbounded sparse potentials particularly with the potentials of the special type which give rise to the spectra with hausdorff dimensionality between 1 2 and 1 operators with bounded sparse potentials are also considered

discretized gabor based beam algorithm for time harmonic radiation from two dimensional truncated planar aperture distributions i formulation and solution in this first part of a two paper sequence we develop a gabor based gaussian beam gb method for the representation of three dimensional 3 d time harmonic vector electromagnetic fields excited by two dimensional 2 d truncated arbitrarily polarized planar aperture field distributions the biorthogonal gabor basis is tied to a lattice in the discretized four dimensional 4 d configuration space spectrum wavenumber phase space which spans the 2 d aperture plane this study generalizes previous investigations of the simpler corresponding procedure for 2 d fields excited by one dimensional 1 d apertures by subsequent specialization in the 1 d aperture case to narrow waisted 2 d ray like gbs we have shown that tracking such beams through interactions with complex environments and recombining them to synthesize the total 2 d field produces robust efficient and accurate algorithms that are useful for a variety of forward and inverse scattering scenarios extension to the time domain via narrow waisted pulsed gbs has likewise been considered these potential applications have motivated the extension here to general 3 d em fields excited by time harmonic 2 d truncated apertures the presentation relates each step in the analytic development to a corresponding step in the 1 d aperture case thereby highlighting the complications in the parameterizing phase space associated with the 2 d aperture problem the outcome is the formal exact solution of the problem under consideration

on the application of some of m g krein s results to the spectral analysis of sturm liouville operators abstract in one of his works m g krein discovered an analogy between polynomials orthogonal on the unit circle and generalized eigenfunctions of certain differential systems we used some ideas of this paper to obtain new results in spectral analysis of sturm liouville operators

absence of singular continuous spectrum for perturbed discrete schrodinger operators abstract we show that the spectral measure of discrete schrodinger operators h u n u n 1 u n 1 v n u n does not have singular continuous component if the potential v n o n 1

orthogonal polynomials with recursion coefficients of generalized bounded variation we consider probability measures on the real line or unit circle with jacobi or verblunsky coefficients satisfying an l p condition and a generalized bounded variation condition this latter condition requires that a sequence can be expressed as a sum of sequences l each of which has rotated bounded variation i e sum n 0 infty vert e i phi l beta n 1 l beta n l vert infty for some phi l this includes a large class of discrete schrodinger operators with almost periodic potentials modulated by l p decay i e linear combinations of lambda n rm cos 2 pi alpha n phi with lambda in ell p of bounded variation and any

power law subordinacy and singular spectra i half line operators we study hausdorff dimensional spectral properties of certain whole line quasiperiodic discrete schrodinger operators by using the extension of the gilbert pearson subordinacy theory that we previously developed in 19

an efficient algorithm to compute the complete set of discrete gabor coefficients the discrete gabor 1946 transform algorithm is introduced that provides an efficient method of calculating the complete set of discrete gabor coefficients of a finite duration discrete signal from finite summations and to reconstruct the original signal exactly from the computed expansion coefficients the similarity of the formulas between the discrete gabor transform and the discrete fourier transform enables one to employ the fft algorithms in the computation the discrete 1 d gabor transform algorithm can be extended to 2 d as well

on subordinacy and analysis of the spectrum of schrodinger operators with two singular endpoints the theory of subordinacy is extended to all one dimensional schrodinger operatorsfor which the corresponding differential expression l d 2 dr 2 v r is in the limit point case at both ends of an interval a b with v r locally integrable this enables a detailed classification of the absolutely continuous and singular spectra to be established in terms of the relative asymptotic behaviour of solutions of lu xu x e as r a and r b the result provides a rigorous but straightforward method of direct spectral analysis which has very general application and somefurther properties of the spectrum are deduced from the underlying theory

positive lyapunov exponents for a class of deterministic potentials letv be a smooth non constant function on the torus and lett be a hyperbolic toral automorphism consider a discrete one dimensional schrodinger operatorh whose potential at sitej is given bygv j gv t j we prove that wheng 0 is small andg 1 2 e 2 g 1 2 the lyapunov exponent for the cocycle generated byh e is proportional tog 2 the proof relies on a formula of pastur and figotin and on symbolic dynamics

schrodinger operators with many bound states consider the schrodinger operators h d2 dx2 v x we present a method for estimating the potential in terms of the negative eigenvalues of these operators among the applications are inverse lieb thirring inequalities and several sharp results concerning the spectral properties of h

narrow waisted gaussian beam discretization for short pulse radiation from one dimensional large apertures we develop a gabor based gaussian beam gb algorithm for representing two dimensional 2 d radiation from finite aperture distributions with short pulse excitation in the time domain td the work extends previous results using 2 d frequency domain fd narrow waisted gaussian beams the fd algorithm evolves from the rigorous kirchhoff integration over the aperture distribution which is then parameterized via the discrete gabor basis and evaluated asymptotically for high frequencies to furnish the gb propagators to the observer the td results are obtained by fourier inversion from the fd and yield pulsed beams pb we describe the resulting td algorithm for several aperture distributions ranging from simple linearly phased linear delay to arbitrary time delay profiles the latter accommodate the important case of focusing td aperture fields for modulated pulses with gaussian envelopes we compute accurate closed form analytic solutions which have been calibrated against numerical reference data our results confirm that the previously established utility of the gabor based narrow waisted fd gb algorithm for radiation from distributed apertures remains intact in the td

singular continuous spectrum is generic in a variety of contexts we prove that singular continuous spectrum is generic in the sense that for certain natural complete metric spaces of operators those with singular spectrum are a dense g delta

de localization in the prime schrodinger operator we report a combined numerical approach to study the localization properties of the one dimensional tight binding model with potential modulated along the prime numbers a localization delocalization transition was found as a function of the potential intensity it is also argued that there are delocalized states for any value of the potential intensity

on consistency of subspace methods for system identification subspace methods for identification of linear time invariant dynamical systems typically consist of two main steps first a so called subspace estimate is constructed this first step usually consists of estimating the range space of the extended observability matrix secondly an estimate of system parametersis obtained based on the subspace estimate in this paper the consistency of a large class of methods for estimating the extended observability matrix is analyzed persistence of excitation conditions on the input signal are given which guarantee consistent estimates for systems with only measurement noise for systems with process noise it is shown that a persistence of excitation condition on the input is not sufficient more precisely an example for which the subspace methods fail to give a consistent estimate of the transfer function is given this failure occurs even if the input is persistently exciting of any order it is also shown that this problem can be eliminated if stronger conditions on the input signal are imposed

identification of dynamic errors in variables models this paper deals with the identifiability of scalar dynamic errors in variables models characterized by rational spectra the hypothesis of causality for the underlying dynamic system is taken into account by making use of stochastic realization theory and of the structural properties of state space representations it is shown that under mild assumptions the model is uniquely identified

finite gap potentials and wkb asymptotics for one dimensional schrodinger operators consider the schrodinger operator h d2 dx2 v x with power decaying potential v x o x we prove that a previously obtained dimensional bound on exceptional sets of the wkb method is sharp in its whole range of validity the construction relies on pointwise bounds on finite gap potentials these bounds are obtained by an analysis of the jacobi inversion problem on hyperelliptic riemann surfaces

subspace algorithms for the identification of multivarible dynamic errors in variables models abstract we consider the problem of identifying multivariable finite dimensional linear time invariant systems from noisy input output measurements apart from the fact that both the measured input and output are corrupted by additive white noise the output may also be contaminated by a term which is caused by a white input process noise furthermore all these noise processes are allowed to be correlated with each other we shall develop a solution to this problem in the framework of subspace identification and we shall show that our algorithms give consistent estimates when the system is operating in open or closed loop two realistic simulation studies are presented to demonstrate the practical applicability of the proposed algorithms 1997 elsevier science ltd

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

pure point spectrum under 1 parameter perturbations and instability of anderson localization we consider a selfadjoint operator a and a selfadjoint rank one projection p onto a vector which is cyclic fora we study the set of all eigenvalues of the operatora t a tp t that belong to its essential spectrum which does not depend on the parametert we prove that this set is empty for a dense set of values oft then we apply this result or its idea to questions of anderson localization for 1 dimensional schrodinger operators discrete and continuous

discretized gabor based beam algorithm for time harmonic radiation from two dimensional truncated planar aperture distributions ii asymptotics and numerical tests for pt i see ibid vol 50 no 12 p 1751 59 2002 in this second part of the two part sequence dealing with gabor based gaussian beam gb representations for the excitation of time harmonic three dimensional 3 d vector electromagnetic fields excited by two dimensional 2 d arbitrarily polarized truncated planar aperture distributions set in a discretized configuration space spectrum wavenumber phase space we employ high frequency asymptotic approximations to reduce the formal solutions developed in part i to efficient algorithms for implementation the resulting explicit expressions for the 3 d gb propagators are applied to the species of narrow waisted gbs which possess ray like features without the failures of ray fields in ray optical transition regions the potential utility of such gbs in the synthesis of wave interactions with complex environments has been discussed previously the narrow waisted gb algorithms for the aperture and radiated near to far zone fields are calibrated for robustness accuracy and efficiency by comparison with numerically generated reference solutions in a series of tests involving coordinate separable rectangular aperture distributions with cosine amplitude tapers

on subordinacy and analysis of the spectrum of one dimensional schrodinger operators abstract by considering the behaviour as n of the ratio of l2 0 n norms of solutions of d 2 dr 2 v r u xu 0 r x r a characterisation of the absolutely continuous and singular spectra of one dimensional schrodinger operators is deduced the analysis is applicable to all operators for which l d 2 dr 2 v r is regular at 0 and in the limit point case at infinity with v r locally integrable

identification methods in a unified framework the paper derives a framework suitable to discuss the classical koopmans levin kl and maximum likelihood ml algorithms to estimate parameters of errors in variables linear models in a unified way using the capability of the unified approach a new parameter estimation algorithm is presented offering flexibility to ensure acceptable variance in the estimated parameters the developed algorithm is based on the application of hankel matrices of variable size and can equally be considered as a generalized version of the kl method gkl or as a reduced version of the ml estimation the methodology applied to derive the gkl algorithm is used to present a straightforward derivation of the subspace identification algorithm

spectral and dynamical properties of certain random jacobi matrices with growing parameters in this paper a family of random jacobi matrices with off diagonal terms that exhibit power law growth is studied since the growth of the randomness is slower than that of these terms it is possible to use methods applied in the study of schr odinger operators with random decaying potentials a particular result of the analysis is the existence of operators with arbitrarily fast transport whose spectral measure is zero dimensional the results are applied to the infinite dumitriu edelman model 2002 and its spectral properties are analyzed

operators with singular continuous spectrum ii rank one operators for an operator a with cyclic vector we studya p wherep is the rank one projection onto multiples of if spec a anda has no a c spectrum we prove thata p has purely singular continuous spectrum on for a denseg of s

time domain radiation from large two dimensional apertures via narrow waisted gaussian beams this paper deals with the short pulse radiation of three dimensional 3 d vector electromagnetic fields from arbitrarily polarized large two dimensional 2 d truncated aperture distributions which are parameterized in terms of narrow waisted ray like pulsed gaussian basis beams centered on a discretized gabor lattice in a four dimensional 4 d configuration spectrum phase space the study extends our previous gabor based investigation of time domain td short pulse radiation of 2 d fields from 1 d large truncated apertures with nonphased linearly phased delayed and nonlinearly phased focusing aperture field profiles we begin with and summarize a gabor based frequency domain fd formulation of the 2 d aperture problem which has been presented and tested elsewhere but we include additional numerical examples for validation and quality assessment as done by galdi felsen and castanon see ibid vol 49 p 1322 32 sept 2001 we access the time domain by fourier inversion from the fd starting from the initial 3 d space time kirchhoff formulation whose numerical integration furnishes reference solutions and then passing on to gabor parameterized field representations in terms of pulsed beam pb wavepackets which are launched by linearly and nonlinearly phase delayed focusing aperture distributions example calculations and comparisons with numerically generated reference data serve to calibrate the gabor pb algorithms and assess their domains of validity

brief paper maximum likelihood identification of noisy input output models this work deals with the identification of errors in variables models corrupted by white and uncorrelated gaussian noises by introducing an auxiliary process it is possible to obtain a maximum likelihood solution of this identification problem by means of a two step iterative algorithm this approach allows also to estimate as a byproduct the noise free input and output sequences moreover an analytic expression of the finite cramer rao lower bound is derived the method does not require any particular assumption on the input process however the ratio of the noise variances is assumed as known the effectiveness of the proposed algorithm has been verified by means of monte carlo simulations

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

critical coupling constants and eigenvalue asymptotics of perturbed periodic sturm liouville operators perturbations of asymptotic decay c r 2 arise in the partial wave analysis of rotationally symmetric partial differential operators we show that for each end point 0 of the spectral bands of a perturbed periodic sturm liouville operator there is a critical coupling constant c crit such that eigenvalues in the spectral gap accumulate at 0 if and only if c c crit 1 the oscillation theoretic method used in the proof also yields the asymptotic distribution of the eigenvalues near 0

bounded solutions and absolute continuity of sturm liouville operators this work is a contribution to the theory of the absolutely continuous spectrum of one dimensional schriidinger operators i e sturm liouville operators generated by differential expressions of the form ru u qu in proofs of absolute continuity for such operators and also for the closely related dirac systems always asymptotic properties of solutions of zu lu for real or complex values of 1 have played an important role see for example 18 19 8 4 9 lo in these works and a number of others particular classes of potentials q were treated using properties of solutions in the course of the proof of absolute continuity somewhat more recently the connection between asymptotic properties of solutions and absolute continuity was given a theory i e results of the following form were proved let the solutions of zu lu satisfy some specific asymptotic property for a varying in some subset of r no or almost no a priori assumptions on q then operators generated by z are absolutely continuous in this set a first result of this type was given by carmona 3 who proved that local uniform boundedness of solutions together with their derivatives in some open a set implies pure absolute continuity in this set here we will mostly rely on the concept of non or racy of solutions which was shown by gilbert and pearson 7 12 61 to be a powerful tool in the investigation of the continuous spectrum of sturm liouville operators in section 2 we give some consequences of their results whi ch are useful in proofs of absolute continuity these consequences range from non existence of imbedded eigenvalues and the existence of absolutely continuous parts to pure absolute continuity and the existence of transient 210 0022 247x 92 5 00

suppression of bursting abstract we investigate the possibility of using a single small amplitude control input and feedback to stabilize equilibrium sets in a class of highly nonlinear o 2 symmetric dynamical systems possessing structurally stable heteroclinic cycles the leading order behavior near the equilibria is bilinear and homogeneous in the state variables while nonlinearities representing the symmetry breaking effect of the controller are crucial after a series of simplifying transformations we use ideas from optimal control theory to construct a stabilizing controller this study is motivated by the desire to stabilize the burst sweep cycle in low dimensional models of a turbulent boundary layer in the last two sections we apply the techniques to the 10 dimensional system of aubry et al 1988 kw bilinear systems dynamical systems heteroclinic cycles normal forms optimal control symmetry

subspace based methods for the identification of linear time invariant systems subspace based methods for system identification have attracted much attention during the past few years this interest is due to the ability of providing accurate state space models for multivariable linear systems directly from input output data the methods have their origin in classical state space realization theory as developed in the 1960s the main computational tools are the qr and the singular value decompositions here an overview of existing subspace based techniques for system identification is given the methods are grouped into the classes of realization based and direct techniques similarities between different algorithms are pointed out and their applicability is commented upon we also discuss some recent ideas for improving and extending the methods a simulation example is included for comparing different algorithms the subspace based approach is found to perform competitive with respect to prediction error methods provided the system is properly excited

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

value distribution and spectral theory of schrodinger operators with l 2 sparse potentials we apply the methods of value distribution theory to the spectral asymptotics of schrodinger operators with l2 sparse potentials

modified prufer and efgp transforms and the spectral analysis of one dimensional schrodinger operators using control of the growth of the transfer matrices wediscuss the spectral analysis of continuum and discrete half line schrodinger operators with slowly decaying potentials among our results we show if where w has compact support and then h has purely a c resp purely s c spectrum on 0 if for n 1 2 a n potentials where a n are independent identically distributed random variables with e a n 0 e a 2 n 1 and 2 we find singular continuous spectrum with explicitly computable fractional hausdorff dimension

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

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

systematic study of fields due to extended apertures by gaussian beam discretization gaussian beams are used as basis elements in field representations to gain insight into how the choice of beam parameters affects the final representation a systematic study for the simple test case of a one dimensional linearly phased cosine aperture distribution has been undertaken by successively adding individual displaced and or tilted beams with large narrow or matched waists one can assess how the elements in various portions of the lattice contribute to the build up of the actual field in the aperture near zone and far zone adding enough beams always guarantees homing in on the exact solution as is verified here by independent comparison different beam choices imply different modeling of the radiation process the understanding gained thereby is helpful for selecting beam parameters in subsequent applications where it is necessary to balance requirements of good convergence ease of computation and ability to track the beams through perturbing environments like a radome indications are that the narrow beams provide the most robust and versatile formulation to deal with these generalized conditions

consistency of system identification by global total least squares global total least squares gtls is a method for the identification of linear systems where no distinction between input and output variables is required this method has been developed within the deterministic behavioural approach to systems in this paper we analyse statistical properties of this method when the observations are generated by a multivariable stationary stochastic process in particular sufficient conditions for the consistency of gtls are derived this means that when the number of observations tends to infinity the identified deterministic system converges to the system that provides an optimal appoximation of the data generating process the two main results are the following gtls is consistent if a guaranteed stability bound can be given a priori if this information is not available then consistency is obtained if gtls is applied to the observed data extended with zero values in past and future

absolutely continuous spectrum of one dimensional schrodinger operators and jacobi matrices with slowly decreasing potentials we prove that for any one dimensional schrodinger operator with potentialv x satisfying decay condition v x cx 3 4 e the absolutely continuous spectrum fills the whole positive semi axis the description of the set in on which the singular part of the spectral measure might be supported is also given analogous results hold for jacobi matrices

sparse one dimensional discrete dirac operators ii spectral properties we study spectral properties of some discrete dirac operators with nonzero potential only at some sparse and suitably randomly distributed positions as observed in the corresponding schrodinger operators we determine the hausdorff dimension of its spectral measure and identify a sharp spectral transition from point to singular continuous

some jacobi matrices with decaying potential and dense point spectrum we discuss doubly infinite matrices of the formm ij i j 1 i j 1 v i ij as operators on l2 we present for each 0 examples of potentialsv n with v n o n 1 2 and wherem has only point spectrum our discussion should be viewed as a remark on the recent work of delyon kunz and souillard

operators with singular continuous spectrum vii examples with borderline time decay we construct one dimensional potentials v x so that if h d 2 dx 2 v x on l 2 then h has purely singular spectrum but for a dense set d d implies that ith c t 1 2 in t for t 2 this implies the spectral measures have hausdorff dimension one and also following an idea of malozemov molchanov provides counterexamples to the direct extension of the theorem of simon spencer on one dimensional infinity high barriers

on the absolutely continuous spectrum of one dimensional schrodinger operators with square summable potentials for continuous and discrete one dimensional schrodinger operators with square summable potentials the absolutely continuous part of the spectrum is essentially supported by 0 and 2 2 respectively this fact is proved by considering a priori estimates for the transmission coefficient

some schrodinger operators with power decaying potentials and pure point spectrum we construct deterministic potentials such that the schrodinger equation on has dense pure point spectrum in for almost all boundary conditions at as a by product we also obtain power decaying potentials for which the spectrum is purely singular continuous on for all boundary conditions