Quantitative Limiting Absorption Principle in the Semiclassical Limit

We give an elementary proof of Burq’s resolvent bounds for long range semiclassical Schrödinger operators. Globally, the resolvent norm grows exponentially in the inverse semiclassical parameter, and near infinity it grows linearly. We also weaken the regularity assumptions on the potential.     

Approximated structured pseudospectra

Pseudospectra and structured pseudospectra are important tools for the analysis of matrices. Their computation, however, can be very demanding for all but small matrices. A new approach to compute approximations of pseudospectra and structured pseudospectra, based on determining the spectra of many suitably chosen rank‐one or projected rank‐one perturbations of the given matrix is proposed. The choice of rank‐one or projected rank‐one perturbations is inspired by Wilkinson's analysis of eigenvalue sensitivity. Numerical examples illustrate that the proposed approach gives much better insight into the pseudospectra and structured pseudospectra than random or structured random rank‐one perturbations with lower computational burden. The latter approach is presently commonly used for the determination of structured pseudospectra.     

Characterization,parity, and power sequences of locally finite doubly stochastic matrices

We show that infinite locally finite doubly stochastic matrices are particular limits of sequences of finite doubly stochastic matrices and reciprocally. Thereby, we define the parity in the set of infinite locally finite doubly stochastic matrices. In particular, convexity and stability properties of the even matrix of this set are investigated, as well as the differences between the finite case and the infinite case. Moreover, the limits of the powers of locally finite infinite doubly stochastic matrices in this context are determined.     

Remarks on the Convergence of Pseudospectra

We establish the convergence of pseudospectra in Hausdorff distance for closed operators acting in different Hilbert spaces and converging in the generalised norm resolvent sense. As an assumption, we exclude the case that the limiting operator has constant resolvent norm on an open set. We extend the class of operators for which it is known that the latter cannot happen by showing that if the resolvent norm is constant on an open set, then this constant is the global minimum. We present a number of examples exhibiting various resolvent norm behaviours and illustrating the applicability of this characterisation compared to known results.     

Pseudospectra of rectangular matrices

Pseudospectra of rectangular matrices vary continuously withthe matrix entries, a feature that eigenvalues of these matricesdo not have. Some properties of eigenvalues and pseudospectraof rectangular matrices are explored, and an efficient algorithmfor the computation of pseudospectra is proposed. Applicationsare given in (square) eigenvalue computation (Lanczos iteration),square pseudospectra approximation (Arnoldi iteration), controltheory (nearest uncontrollable system) and game theory.     

Pseudospectra and delay differential equations

In this paper, we present a new method for computing the pseudospectra of delay differential equations (DDEs) with fixed finite delay. This provides information on the sensitivity of eigenvalues under arbitrary perturbations of a given size, and hence insight into how stability may change under variation of parameters. We also investigate how differently weighted perturbations applied to the individual matrices of the delayed eigenvalue problem affect the pseudospectra. Furthermore, we compute pseudospectra of the infinitesimal generator of the DDE, from which a lower bound on the maximum transient growth can be inferred. To illustrate our method, we consider a DDE modelling a semiconductor laser subject to external feedback.     

Accurate computations of matrices with bidiagonal decomposition using methods for totally positive matrices

A class of sign‐symmetric P‐matrices including all nonsingular totally positive matrices and their inverses as well as tridiagonal nonsingular H‐matrices is presented and analyzed. These matrices present a bidiagonal decomposition that can be used to obtain algorithms to compute with high relative accuracy their singular values, eigenvalues, inverses, or their LDU factorization. Copyright © 2012 John Wiley & Sons, Ltd.     

Pseudospectra do not determine norm behavior, even for matrices with only simple eigenvalues

We present an example of a pair of 4×4 matrices having identical pseudospectra but whose squares have different norms. The novelty of the example lies in the fact that the matrices in question have only simple eigenvalues.     

Word functions of pseudo-Markov chains

A word function is a function from the set of all words over a finite alphabet into the set of real numbers. In particular, when the blocks of a partition over the state set of a Markov chain are taken as the letters of the finite alphabet, and the function represents the probabilities that the chain will visit sequences of such blocks consecutively, then the function is a function of a Markov chain. It is known that (the rank of a function is defined in the text), a word function is of “finite rank” if and only if it is a function of a pseudo Markov chain (“pseudo” means here that the initial vector and the matrix representing the chain may have positive, negative, or zero values and are not necessarily stochastic). The aim of this note is to show that any function of a pseudo Markov chain can be represented as the difference of two functions of true Markov chains multiplied by a factor which grows exponentially with the length of the arguments (considered as words over a finite alphabet).     

A note on the existence of stochastic integro‐differential equations with memory

In this study, we investigate the existence of mild solutions for a class of impulsive neutral stochastic integro‐differential equations with infinite delays, using the Krasnoselskii–Schaefer type fixed point theorem combined with theories of resolvent operators. As an application, an example is provided to illustrate the obtained result. Copyright © 2014 John Wiley & Sons, Ltd.     

Kotani–Last problem and Hardy spaces on surfaces of Widom type

This is a small theory of non almost periodic ergodic families of Jacobi matrices with purely (however) absolutely continuous spectrum. The reason why this effect may happen is that under our “axioms” we found an analytic condition on the resolvent set that is responsible for (exactly equivalent to) this effect.     

Convolution equations and linear systems

This paper developes further the connections between linear systems and convolution equations. Here the emphasis is on equations on finite intervals. For these equations a new characteristic matrix (or operator) function is introduced, which contains all the important information about the equations and the corresponding operators. Explicit formulas for solutions and resolvent kernels are obtained. Convolution equations on the full line are also analyzed. Analogous results are derived for the inversion of finite and infinite Toeplitz matrices.     

Upper packing dimension of a measure and the limit distribution of products of i.i.d. stochastic matrices

This article gives sufficient conditions for the limit distribution of products of i.i.d. 2 × 2 stochastic matrices to be continuous singular, when the support of the distribution of the individual random matrices is countably infinite. It extends a previous result for which the support of the random matrices is finite. The result is based on adapting existing proofs in the context of attractors and iterated function systems to the case of infinite iterated function systems.     

Ideas in the theory of random media

These lectures discuss the ideas of localization, intermittency, and random fluctuations in the theory of random media. These ideas are compared and contrasted with the older approach based on averaging. Within this framework, the topics discussed include: Anderson localization, turbulent diffusion and flows, periodic Schrödinger operators and averaging theory, longwave oscillations of elastic random media, stochastic differential equations, the spectral theory of Hamiltonians with (an infinite sequence of) wells, random Schrödinger operators, electrons in a random homogeneous field, influence of localization effects on the propagation of elastic waves, the Lyapunov spectrum (Lyapunov exponents), the Furstenberg and Oseledec theorems for ann-tuple of identically distributed unimodular matrices and their relation with the spectral theory of random Schrödinger or string operators, Rossby waves, averaging on random Schrödinger operators, percolation mechanisms, the moments method in the theory of sequences of random variables, the evolution of a magnetic field in the turbulent flow of a conducting fluid or plasma (the so-called kinematical dynamo problem), heat transmission in a randomly flowing fluid.     

基于列随机矩阵的逐次差分代换与正半定型的机械化判定

本文选择列随机平均矩阵T_n作为基本代换矩阵,建立了基于T_n的逐次差分代换方法.获得了R_+~n上正半定型,不定型判定的充要条件.并进一步证明了:正定型的差分代换集序列正向终止.根据这些结果编写的Maple程序TSDS3,能够自动证明代数型不等式,对不成立的不等式总能输出反例.该程序虽可能不停机,但大量的应用实例证实了该方法的实用性.     

Accurate computations with collocation matrices of a general class of bases

In this paper, we provide algorithms for computing the bidiagonal decomposition of the collocation matrices of a very general class of bases of interest in computer‐aided geometric design and approximation theory. It is also shown that these algorithms can be used to perform accurately some algebraic computations with these matrices, such as the calculation of their inverses, their eigenvalues, or their singular values. Numerical experiments illustrate the results.     

Bounds on widths of resonances for stark hamiltonians

Resonances of Schrödinger operators are closely related to the poles of certain meromorphic extension of the resolvent. In this paper, we study the resonances in Stark effect, generated by negative eigenvalues. We give precise location for the resonances and prove in particular that the widths of resonances are exponentially small and can be measured by some Agmon's distance.     

A Jacobi–Davidson type method with a correction equation tailored for integral operators

We propose two iterative numerical methods for eigenvalue computations of large dimensional problems arising from finite approximations of integral operators, and describe their parallel implementation. A matrix representation of the problem on a space of moderate dimension, defined from an infinite dimensional one, is computed along with its eigenpairs. These are taken as initial approximations and iteratively refined, by means of a correction equation based on the reduced resolvent operator and performed on the moderate size space, to enhance their quality. Each refinement step requires the prolongation of the correction equation solution back to a higher dimensional space, defined from the infinite dimensional one. This approach is particularly adapted for the computation of eigenpair approximations of integral operators, where prolongation and restriction matrices can be easily built making a bridge between coarser and finer discretizations. We propose two methods that apply a Jacobi–Davidson like correction: Multipower Defect-Correction (MPDC), which uses a single-vector scheme, if the eigenvalues to refine are simple, and Rayleigh–Ritz Defect-Correction (RRDC), which is based on a projection onto an expanding subspace. Their main advantage lies in the fact that the correction equation is performed on a smaller space while for general solvers it is done on the higher dimensional one. We discuss implementation and parallelization details, using the PETSc and SLEPc packages. Also, numerical results on an astrophysics application, whose mathematical model involves a weakly singular integral operator, are presented.     

Exponential convergence of products of stochastic matrices

This paper considers a finite set of stochastic matrices of finite order. Conditions are given under which any product of matrices from this set converges to a constant stochastic matrix. Also, it is shown that the convergence is exponentially fast.     

Computing the field of values and pseudospectra using the Lanczos method with continuation

The field of values and pseudospectra are useful tools for understanding the behaviour of various matrix processes. To compute these subsets of the complex plane it is necessary to estimate one or two eigenvalues of a large number of parametrized Hermitian matrices; these computations are prohibitively expensive for large, possibly sparse, matrices, if done by use of the QR algorithm. We describe an approach based on the Lanczos method with selective reorthogonalization and Chebyshev acceleration that, when combined with continuation and a shift and invert technique, enables efficient and reliable computation of the field of values and pseudospectra for large matrices. The idea of using the Lanczos method with continuation to compute pseudospectra is not new, but in experiments reported here our algorithm is faster and more accurate than existing algorithms of this type.This work was supported by Engineering and Physical Sciences Research Council grants GR/H/52139 and GR/H/94528.