band structure of absorptive photonic crystals the band structure of absorptive dielectric photonic crystals is investigated provided the frequency dependent electric permeability e x satisfies certain analyticity requirements as a function of frequency we show that no bandgaps exist in frequency regions where absorption takes place i e where e x has a non zero imaginary part in this case real eigenvalues of the helmholtz operator in the bloch decomposed formalism are absent using a suitable analytic continuation procedure we find that the former change into resonances i e complex numbers depending on k the wavevector from the first brillouin zone thus leading to complex bands in the lower half plane this is confirmed numerically for a simple one dimensional example

bound states of a one band model for 3d periodic medium we consider the bound states generated by impurities in a periodic three dimensional medium the computations of eigenstates and eigenfunctions for a one band model of the medium are carried out using numerical evaluation of the green s function for bound eigenmode representations we achieve high accuracy in the computation of the eigenmodes and eigenstates for a variety of defects for appropriately adjusted models these computations can be useful for numerical estimates of the minimal configuration of defects capable of introducing a bound state in a gap as well as estimating the rate of the exponential decay of the bound state

localization phenomenon in gaps of the spectrum of random lattice operators we consider a class of random lattice operators including schrodinger operators of the formh w gv wherew x is a real valued periodic function g is a positive constant andv x x d are independent identically distributed real random variables we prove that if the operator w has gaps in the spectrum andg is sufficiently small then the operatorh develops pure point spectrum with exponentially decaying eigenfunctions in a vicinity of the gaps

an optimal wegner estimate and its application to the global continuity of the integrated density of states for random schrodinger operators we prove that the integrated density of states ids of random schrodinger operators with anderson type potentials on l 2 r d for d geq1 is locally holder continuous at all energies with the same holder exponent 0 alpha leq1 as the conditional probability measure for the single site random variable as a special case we prove that if the probability distribution is absolutely continuous with respect to lebesgue measure with a bounded density then the ids is lipschitz continuous at all energies the single site potential u in l 0 infty r d must be nonnegative and compactly supported the unperturbed hamiltonian must be periodic and satisfy a unique continuation principle we also prove analogous continuity results for the ids of random anderson type perturbations of the landau hamiltonian in two dimensions all of these results follow from a new wegner estimate for local random hamiltonians with rather general probability measures

linear prediction methods for blind fractionally spaced equalization we describe adaptive methods for estimating fir zero forcing blind equalizers with arbitrary delay directly from the linear predictions of the observations while most current methods require inversion or singular value decomposition svd of the correlation matrix our methods need only to solve two linear prediction problems they can be implemented as rls or lms algorithms to recursively update the equalizer estimation they are computationally efficient the computational complexity in each recursion can be less than 15 ln sup 2 in the rls case where ln equals the equalizer length and 3l ln in the lms case where l is the number of subchannels the performance of the proposed methods and comparisons with existing approaches are shown by simulation to demonstrate their effectiveness

new characterizations of the region of complete localization for random schrodinger operators we study the region of complete localization in a class of random operators which includes random schrodinger operators with anderson type potentials and classical wave operators in random media as well as the anderson tight binding model we establish new characterizations or criteria for this region of complete localization given either by the decay of eigenfunction correlations or by the decay of fermi projections these are necessary and sufficient conditions for the random operator to exhibit complete localization in this energy region using the first type of characterization we prove that in the region of complete localization the random operator has eigenvalues with finite multiplicity

qr factorization based blind channel identification with second order statistics most eigenstructure based blind channel identification and equalization algorithms with second order statistics need svd or evd of the correlation matrix of the received signal in this paper we address new algorithms based on qr factorization of the received signal directly without calculating the correlation matrix this renders the qr factorization based algorithms more robust against ill conditioned channels i e those channels with almost common zeros among the subchannels first we present a block algorithm that performs the qr factorization of the received data matrix as a whole then a recursive algorithm is developed based on the qr factorization by updating a rank revealing ulv decomposition compared with existing algorithms in the same category our algorithms are computationally more efficient the computation in each recursion of the recursive algorithm is on the order of o m sup 2 if only equalization is required where m is the dimension of the received signal vector our recursive algorithm preserves the fast convergence property of the subspace algorithms thus converging faster than other adaptive algorithms such as the super exponential algorithm with comparable computational complexities moreover our proposed algorithms do not require noise variance estimation numerical simulations demonstrate the good performance of the proposed algorithms

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

a new proof of localization in the anderson tight binding model we give a new proof of exponential localization in the anderson tight binding model which uses many ideas of the frohlich martinelli scoppola and spencer proof but is technically simpler particularly the probabilistic estimates

generalized eigenfunctions for waves in inhomogeneous media abstract many wave propagation phenomena in classical physics are governed by equations that can be recast in schrodinger form in this approach the classical wave equation e g maxwell s equations acoustic equation elastic equation is rewritten in schrodinger form leading to the study of the spectral theory of its classical wave operator a self adjoint partial differential operator on a hilbert space of vector valued square integrable functions physically interesting inhomogeneous media give rise to nonsmooth coefficients we construct a generalized eigenfunction expansion for classical wave operators with nonsmooth coefficients our construction yields polynomially bounded generalized eigenfunctions the set of generalized eigenvalues forming a subset of the operator s spectrum with full spectral measure

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

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

on the birman schwinger principle applied to δ m2 m the condition for e 0 to be an eigenvalue of the operator m2 m v is obtained through the use of the birman schwinger principle theorem 3 2 by setting e 2 and using the analyticity of the corresponding birman schwinger kernel theorem 3 1 the series development of 1 is obtained up to second order on theorem 4 1

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

sampling and equidistribution theorems for elliptic second order operators lifting of eigenvalues and applications abstract we consider elliptic second order partial differential operators with lipschitz continuous leading order coefficients on finite cubes and the whole euclidean space we prove quantitative sampling and equidistribution theorems for eigenfunctions the estimates are scale free in the sense that for a sequence of growing cubes we obtain uniform estimates these results are applied to prove lifting of eigenvalues as well as the infimum of the essential spectrum and an uncertainty relation aka spectral inequality for short energy interval spectral projectors several applications including random operators are discussed in the proof we have to overcome several challenges posed by the variable coefficients of the leading term

on guided waves in 3 d photonic crystals the propagation of guided waves in 3 d photonic crystals is studied in this paper we suppose that the spectrum of reference medium a perfectly 3 d photonic crystal has band gaps if a line defect is introduced along one of the lattice vectors we prove that spectrum can be created inside the gaps of the reference spectrum if the line defect is sufficiently strong furthermore we also prove that the generalized eigenfunctions corresponding to the generalized eigenvalues created by line defect are exponentially decaying away from the line defect

2 dimensional localization of acoustic waves in random perturbation of periodic media abstract in this work we study internal band edges localization of acoustic waves in 2 dimensional space obtained by random perturbation of some periodic media our results rely on the study of lifshitz tails for the integrated density of states for random acoustic operators of the form a 1 localization is then deduced by the standard multiscale argument

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

prediction error method for second order blind identification blind channel identification methods based on the oversampled channel output are a problem of current theoretical and practical interest in this paper we introduce a second order blind identification technique based on a linear prediction approach in contrast to eigenstructure based methods it will be shown that the linear prediction error method is robust to order overdetermination an asymptotic performance analysis of the proposed estimation method is carried out consistency and asymptotic normality of the estimates is established a closed form expression for the asymptotic covariance of the estimates is given numerical simulations and investigations are finally presented to demonstrate the potential and the robustness of the proposed method

a subspace algorithm for certain blind identification problems the problem of blind identification of p inputs q outputs fir transfer functions is addressed existing subspace identification methods derived for p 1 are first reformulated in particular the links between the noise subspace of a certain covariance matrix of the output signals on which subspace methods build on and certain rational subspaces associated with the transfer function to be identified are elucidated based on these relations we study the behavior of the subspace method in the case where the order of the transfer function is overestimated next an asymptotic performance analysis of this estimation method is carried out consistency and asymptotical normality of the estimates is established a closed form expression for the asymptotic covariance of the estimates is given numerical simulations and investigations are presented to demonstrate the potential of the subspace method finally we take advantage of our new reformulation to discuss the extension of the subspace method to the case p 1 we show where the difficulties lie and we briefly indicate how to solve the corresponding problems the possible connections with classical approaches for ma model estimations are also outlined

localization of classical waves ii electromagnetic waves we consider electromagnetic waves in a medium described by a position dependent dielectric constant we assume that is a random perturbation of a periodic function and that the periodic maxwell operator has a gap in the spectrum where we prove the existence of localized waves i e finite energy solutions of maxwell s equations with the property that almost all of the wave s energy remains in a fixed bounded region of space at all times localization of electromagnetic waves is a consequence of anderson localization for the self adjoint operators we prove that in the random medium described by the random operator exhibits anderson localization inside the gap in the spectrum of this is shown even in situations when the gap is totally filled by the spectrum of the random operator we can prescribe random environments that ensure localization in almost the whole gap

noniterative subspace tracking a rank one spherical subspace update that is appropriate for subspace based methods like music and minimum norm is introduced this noniterative highly parallel numerically stabilized subspace update is closely related to rank one eigenstructure updating however a rank one subspace update involves less computation than simple rank one correlation accumulation moreover the frequency tracking capabilities of the noniterative subspace update are virtually identical to and in some case more robust than the more computationally expensive eigen based methods

eigenvalue branches of the perturbed maxwell operator m λd in a gap of σ m the propagation of guided waves in photonic crystal fibers pcfs is studied a pcf can be regarded as a perfectly two dimensional photonic crystal with a line defect along the axial direction this problem can be treated as an eigenvalue problem for a family of noncompact self adjoint operators under the assumption that the background spectrum has a gap we prove that a line defect can create an eigenvalue of any given fixed value in the gap provided that the defect is strong enough based on a decoupling of regions in r2 by means of dirichlet and neumann boundaries then using the trace ideal estimates we study asymptotic distribution of eigenvalues and bounds on the number of eigenvalue branches in particular we show that if the defect is weak enough no eigenvalues can be created inside the gap

the lp theory of the spectral shift function the wegner estimate and the integrated density of states for some random operators we develop the l p theory of the spectral shift function for p 1 applicable to pairs of self adjoint operators whose difference is in the trace ideal p for 0 p 1 this result is a key ingredient of a new short proof of the wegner estimate applicable to a wide variety of additive and multiplicative random perturbations of deterministic background operators the proof yields the correct volume dependence of the upper bound this implies the local holder continuity of the integrated density of states at energies in the unperturbed spectral gap under an additional condition of the single site potential local holder continuity is proved at all energies this new wegner estimate together with other standard results establishes exponential localization for a new family of models for additive and multiplicative perturbations

two algorithms for fast approximate subspace tracking new fast algorithms are presented for tracking singular values singular vectors and the dimension of a signal subspace through an overlapping sequence of data matrices the basic algorithm is called fast approximate subspace tracking fast the algorithm is derived for the special case in which the matrix is changed by deleting the oldest column shifting the remaining columns to the left and adding a new column on the right a second algorithm fast2 is specified by modifying fast to trade reduced accuracy for higher speed the speed and accuracy are compared with the pl algorithm the past and pastd algorithms and the fst algorithm an extension to multicolumn updates for the fast algorithm is also discussed

the discrete spectrum of the periodic maxwell operator perturbed by a decreasing potential the propagation of electromagnetic waves in dielectric medium with decreasing perturbations defects is studied we use the maxwell operator m p x and the perturbed operator m p x q x to describe the background medium and perturbed medium respectively where 0 is the coupling constant under the assumption that m has a spectral gap g we prove that a defect can create an eigenvalue of any given fixed value g for some 0 provided that the support of the defect is large enough we also consider the asymptotic distribution of eigenvalues in the gap and bounds on the number of eigenvalue branches

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

nonequilibrium steady states of matrix product form a solver s guide we consider the general problem of determining the steady state of stochastic nonequilibrium systems such as those that have been used to model among other things biological transport and traffic flow we begin with a broad overview of this class of driven diffusive systems which includes exclusion processes focusing on interesting physical properties such as shocks and phase transitions we then turn our attention specifically to those models for which the exact distribution of microstates in the steady state can be expressed in a matrix product form in addition to a gentle introduction to this matrix product approach how it works and how it relates to similar constructions that arise in other physical contexts we present a unified pedagogical account of the various means by which the statistical mechanical calculations of macroscopic physical quantities are actually performed we also review a number of more advanced topics including nonequilibrium free energy functionals the classification of exclusion processes involving multiple particle species existence proofs of a matrix product state for a given model and more complicated variants of the matrix product state that allow various types of parallel dynamics to be handled we conclude with a brief discussion of open problems for future research

mathematical analysis of the guided waves in photonic crystal fibers abstract the propagation of guided waves in photonic crystal fibers pcfs is studied the structure of a pcf can be regarded as a perfect two dimensional photonic crystal with a line defect along the invariant direction this problem can be treated as an eigenvalue problem for a family of noncompact self adjoint operators we prove that line defects do not change the essential spectrum of the associated background medium this result plays a key role for studying the influence of line defects on the background spectrum a modified combes thomas estimate is also formulated

nonlinear maxwell equations in inhomogeneous media we study the basic properties of the maxwell equations for nonlinear inhomogeneous media assuming the classical nonlinear optics representation for the nonlinear polarization as a power series we show that the solution exists and is unique in an appropriate space if the excitation current is not too large the solution to the nonlinear maxwell equations is represented as a power series in terms of the solution of the corresponding linear maxwell equations this representation holds at least for the time period inversely proportional to the appropriate norm of the solution to the linear maxwell equation we derive recursive formulas for the terms of the power series for the solution including an explicit formula for the first significant term attributed to the nonlinearity

localization for some continuous random hamiltonians in d dimensions abstract we prove the existence with probability one of an interval of pure point spectrum for some families of continuous random schrodinger operators in d dimensions for anderson like models with positive short range single site potentials we also prove that the corresponding eigenfunctions decay exponentially and that the integrated density of states is lipschitz continuous for the other families of random potentials that we study we show that the corresponding eigenfunctions decay faster than an inverse power of x which depends upon the decay rate of the single site potential to obtain these results we develop an extension of the classical aronszajn donoghue theory for a class of relatively compact perturbations and a spectral averaging method which extends kotani s trick to these more general families of operators

the absolute continuity of the spectrum of maxwell operator in a periodic media we prove that the spectrum of maxwell operator b e rot e a x rot b x b with non negative and periodic functions a x and b x is absolutely continuous

localization of classical waves i acoustic waves we consider classical acoustic waves in a medium described by a position dependent mass density x we assume that x is a reandom perturbation of a periodic function 0 x and that the periodic acoustic operator a 0 nabla cdot tfrac 1 varrho 0 x nabla has a gap in the spectrum we prove the existence of localized waves i e finite energy solutions of the acoustic equations with the property that almost all of the wave s energy remains in a fixed bounded region of space at all times with probability one localization of acoustic waves is a consequence of anderson localization for the self adjoint operators a nabla cdot tfrac 1 varrho 0 x nabla onl2 d we prove that in the random medium described by x the random operatora exhibits anderson localization inside the gap in the spectrum ofa0 this is shown even in situations when the gap is totally filled by the spectrum of the random opertor we can prescribe random environments that ensure localization in almost the whole gap

on the spectrum of schrodinger operators with a random potential we investigate the spectrum of schrodinger operatorsh of the type h qi f x xi i qi and i independent identically distributed random variables i d we establish a strong connection between the spectrum ofh and the spectra of deterministic periodic schrodinger operators from this we derive a condition for the existence of forbidden zones in the spectrum ofh for random one and three dimensional kronig penney potentials the spectrum is given explicitly

the integrated density of states for some random operators with nonsign definite potentials we study the integrated density of states of random anderson type additive and multiplicative perturbations of deterministic background operators for which the single site potential does not have a fixed sign our main result states that under a suitable assumption on the regularity of the random variables the integrated density of states of such random operators is locally holder continuous at energies below the bottom of the essential spectrum of the background operator for any nonzero disorder and at energies in the unperturbed spectral gaps provided the randomness is sufficiently small the result is based on a proof of a wegner estimate with the correct volume dependence the proof relies upon the lp theory of the spectral shift function for p 1 comm math phys 218 2001 113 130 and the vector field methods of klopp comm math phys 167 1995 553 569 we discuss the application of this result to schrodinger operators with random magnetic fields and to band edge localization

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

performance analysis of an adaptive algorithm for tracking dominant subspaces this paper provides a performance analysis of a least mean square lms dominant invariant subspace algorithm based on an unconstrained minimization problem this algorithm is a stochastic gradient algorithm driving the columns of a matrix w to an orthonormal basis of a dominant invariant subspace of a correlation matrix we consider the stochastic algorithm governing the evolution of ww sup h to the projection matrix onto this dominant invariant subspace and study its asymptotic distribution a closed form expression of its asymptotic covariance is given in the case of independent observations and is further analyzed to provide some insights into the behavior of this lms type algorithm in particular it is shown that even though the algorithm does not constrain w to have orthonormal columns there is deviation from orthonormality of the first order we also give a closed form expression of the asymptotic covariance of the doas estimated by the music algorithm applied to the adaptive estimate of the projector it is found that the asymptotic distributions have a structure that is very similar to those describing batch estimation techniques because both algorithms are obtained from the minimization of the same criterion finally the accuracy of the asymptotic analysis is checked by numerical simulations and is found to be valid not only for a small step size but in a very large domain

on updating signal subspaces the authors develop an algorithm for adaptively estimating the noise subspace of a data matrix as is required in signal processing applications employing the signal subspace approach the noise subspace is estimated using a rank revealing qr factorization instead of the more expensive singular value or eigenvalue decompositions using incremental condition estimation to monitor the smallest singular values of triangular matrices the authors can update the rank revealing triangular factorization inexpensively when new rows are added and old rows are deleted experiments demonstrate that the new approach usually requires o n sup 2 work to update an n n matrix and that it accurately tracks the noise subspace

on estimating the condition of eigenvalues and eigenvectors abstract a method is developed for estimating the accuracy of computed eigenvalues and eigenvectors that are obtained via certain eispack subroutines it does this at a cost of o n 2 flops per eigenpair assuming that the eigenpair is known and assuming that the original matrix has been reduced to hessenberg form the heart of the technique involves estimating the smallest singular value of a certain nearly triangular submatrix this is accomplished by some standard zero chasing with givens transformations and with a 2 norm version of the linpack condition estimator an eispack compatible code has been developed and its performance is discussed

blind channel identification subspace tracking method without rank estimation subspace ss methods are an effective approach for blind channel identification however these methods also have two major disadvantages 1 they require accurate channel length estimation and or rank estimation of the correlation matrix which is difficult with noisy channels and 2 they require a large amount of computation for the singular value decomposition svd which makes it inconvenient for adaptive implementation although many adaptive subspace tracking algorithms can be applied the computational complexity is still o m sup 3 where m is the data vector length in this paper we introduce new recursive subspace algorithms using ulv updating and successive cancellation techniques the new algorithms do not need to estimate the rank of the correlation matrix furthermore the channel length can be overestimated initially and be recovered at the end by a successive cancellation procedure which leads to more convenient implementations the adaptive algorithm has computations of o m sup 2 in each recursion the new methods can be applied to either the single user or the multiuser cases simulations demonstrate their good performance

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