Stability of spectral types for Jacobi matrices under decaying random perturbations

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.     

On spectral variations under relatively bounded perturbations

Linear operators in Banach and Hilbert spaces are considered. Bounds for the spectrum are established under relatively bounded perturbations. An application to nonselfadjoint differential operators is discussed.     

Stability of distributed parameter systems with finite-dimensional controller-compensators using singular perturbations

Using a singular perturbation formulation of the linear time-invariant distributed parameter system, we develop a method to design finite-dimensional feedback compensators of any fixed order which will stabilize the infinite-dimensional distributed parameter system. The synthesis conditions are given entirely in terms of a finite-dimensional reduced-order model; the stability results depend on an infinite-dimensional version of the Klimushchev-Krasovskii lemma presented here.     

Coercive singular perturbations

Summary We consider general boundary value problems with small parameter ɛ in the operator and boundary conditions. Both the perturbed and reduced operators are supposed to be elliptic. We point outnecessary andsufficient conditions of Shapiro-Lopatinsky type for the singularly perturbed problem to be coercive, i.e. for a two-sided a priori estimate to hold for its solutions uniformly with respect to ɛ. Entrata in Redazione il 6 luglio 1977.     

Point spectrum and mixed spectral types for rank one perturbations

We consider examples of rank one perturbations with a cyclic vector for . We prove that for any bounded measurable set , an interval, there exist so that
eigenvalue agrees with up to sets of Lebesgue measure zero. We also show that there exist examples where has a.c. spectrum for all , and for sets of 's of positive Lebesgue measure, also has point spectrum in , and for a set of 's of positive Lebesgue measure, also has singular continuous spectrum in .

     

Stability of essential spectra of singular Sturm‐Liouville differential operators under perturbations small at infinity

This paper is concerned with the stability of essential spectra of singular Sturm‐Liouville differential operators with complex‐valued coefficients. It is proved that the essential spectrum of the corresponding minimal operator is preserved by perturbations small at infinity with respect to the unperturbed operator. Based on it, 1‐dimensional Schrödinger operators under local dilative perturbations are studied.     

Stability of generalized equations under nonlinear perturbations

This paper studies solution stability of generalized equations over polyhedral convex sets. An exact formula for computing the Mordukhovich coderivative of normal cone operators to nonlinearly perturbed polyhedral convex sets is established based on a chain rule for the partial second-order subdifferential. This formula leads to a sufficient condition for the local Lipschitz-like property of the solution maps of the generalized equations under nonlinear perturbations.     

On spectral variations under bounded real matrix perturbations

Summary In this paper we investigate the set of eigenvalues of a perturbed matrix {ie509-1} whereA is given and ^{n × n}, ||< is arbitrary. We determine a lower bound for thisspectral value set which is exact for normal matricesA with well separated eigenvalues. We also investigate the behaviour of the spectral value set under similarity transformations. The results are then applied tostability radii which measure the distance of a matrixA from the set of matrices having at least one eigenvalue in a given closed instability domain _b.     

An example of stability of singular spectrum under smooth perturbations

Random Fourier series with Gaussian coefficients and very large gaps are used to construct a class of continuous stationary Gaussian processes whose sample functions X(t) have remarkable properties. Almost surely the function X(t) has an essential range of Lebesgue measure zero and for each smooth function Y(t) the multiplication operator (M_x+yf) (t) [X(t)+Y(t)]f(t) on L²[0,2] has only singular continuous spectrum.     

Upper semicontinuity of attractors for lattice systems under singular perturbations

Consider the following first order lattice system

     

Elliptic difference singular perturbations

Summary Difference approximations for differential singular perturbations with small parameter ɛ are considered. We point out ellipticity and coerciveness conditions which arenecessary andsufficient for a two-sided a priori estimate to hold for the solution of difference singular perturbation uniformly with respect to the ratio of both small parameters: the original one ɛ and the meshsize h. Entrata in Redazione il 27 maggio 1978.     

Supersymmetry and singular perturbations

First it is shown in an abstract framework that supersymmetric quantum theory may give rise to very singular perturbations, which make perturbation theories invalid. Then, the abstract results are applied to a model of 1-dimensional supersymmetric quantum mechanics (Witten's model) to obtain some families of singular Hamiltonians. Some concrete examples, which reveal their singularness, are discussed in detail.     

Smooth approximation of singular perturbations

We consider a certain subclass of self-adjoint extensions of the symmetric operator

     

Stability of large-scale discrete systems under structural perturbations

By using the matrix Lyapunov function, we establish conditions of (uniform) stability and (uniform) asymptotic stability of a large-scale discrete system under structural perturbations.     

Stability of Martin boundary under non-local Feynman-Kac perturbations

Recently the authors showed that the Martin boundary and the minimal Martin boundary for a censored (or resurrected) -stable process Y in a bounded C^1,1-open set D with (1,2) can all be identified with the Euclidean boundary D of D. Under the gaugeability assumption, we show that the Martin boundary and the minimal Martin boundary for the Schrödinger operator obtained from Y through a non-local Feynman-Kac transform can all be identified with D. In other words, the Martin boundary and the minimal Martin boundary are stable under non-local Feynman-Kac perturbations. Moreover, an integral representation of nonnegative excessive functions for the Schrödinger operator is explicitly given. These results in fact hold for a large class of strong Markov processes, as are illustrated in the last section of this paper. As an application, the Martin boundary for censored relativistic stable processes in bounded C^1,1-smooth open sets is studied in detail.This research is supported in part by NSF Grant DMS-0071486 and a RRF Grant from University of WashingtonMathematics Subject Classification (2000):Primary 31C35, 60J45, 35J10; Secondary 60J50, 60J57     

Stability of linear stochastic delay under steady perturbations

Kiev. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 33, No. 5, pp. 107–114, September–October, 1992.