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.     

Absolutely continuous spectrum of one random elliptic operator

We consider an elliptic random operator, which is the sum of the differential part and the potential. The potential considered in the paper is the same as the one in the Andersson model, however the differential part of the operator is different from the Laplace operator. We prove that such an operator has absolutely continuous spectrum on all of (0,∞).     

Absolutely continuous spectrum of Schrödinger operators with potentials slowly decaying inside a cone

For a large class of multi-dimensional Schrödinger operators it is shown that the absolutely continuous spectrum is essentially supported by [0,∞). We require slow decay and mildly oscillatory behavior of the potential in a cone and can allow for arbitrary non-negative bounded potential outside the cone. In particular, we do not require the existence of wave operators. The result and method of proof extends previous work by Laptev, Naboko and Safronov.     

Smoothness of density of states for random decaying interaction

In this paper we consider one dimensional random Jacobi operators with decaying independent randomness and show that under some condition on the decay vis-a-vis the distribution of randomness, that the distribution function of the average spectral measures of the associated operators are smooth.     

Mourre's theory for some unbounded Jacobi matrices

We show a Mourre estimate for a class of unbounded Jacobi matrices. In particular, we deduce the absolute continuity of the spectrum of such matrices. We further conclude some propagation theorems for them.     

On the spectrum of magnetic Dirac operators with Coulomb-type perturbations

We carry out the spectral analysis of singular matrix valued perturbations of 3-dimensional Dirac operators with variable magnetic field of constant direction. Under suitable assumptions on the magnetic field and on the perturbations, we obtain a limiting absorption principle, we prove the absence of singular continuous spectrum in certain intervals and state properties of the point spectrum. Constant, periodic as well as diverging magnetic fields are covered, and Coulomb potentials up to the physical nuclear charge Z<137 are allowed. The importance of an internal-type operator (a 2-dimensional Dirac operator) is also revealed in our study. The proofs rely on commutator methods.     

On the absolutely continuous spectrum of block operator matrices

We prove an abstract theorem on the preservation of the absolutely continuous spectrum for block operator matrices. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the absolutely continuous spectrum of the Laplace–Beltrami operator acting on p ‐forms for a class of warped product metrics

We explicitely compute the absolutely continuous spectrum of the Laplace–Beltrami operator for p ‐forms for the class of warped product metrics dσ ² = y ^2a dy ² + y ^2b dθ ²equation/tex2gif-inf-1.gif, where y is a boundary defining function on the unit ball B (0, 1) in ?^N . (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Attainable densities for random maps

We consider a random map T=T(Γ,ω), where Γ=(τ₁,τ₂,…,τK) is a collection of maps of an interval and ω=(p₁,p₂,…,pK) is a collection of the corresponding position dependent probabilities, that is, pk(x)?0 for k=1,2,…,K and . At each step, the random map T moves the point x to τk(x) with probability pk(x). For a fixed collection of maps Γ, T can have many different invariant probability density functions, depending on the choice of the (weighting) probabilities ω. Most of the results in this paper concern random maps where Γ is a family of piecewise linear semi-Markov maps. We investigate properties of the set of invariant probability density functions of T that are attainable by allowing the probabilities in ω to vary in a certain class of functions. We prove that the set of all attainable densities can be determined algorithmically. We also study the duality between random maps generated by transformations and random maps constructed from a collection of their inverse branches. Such representation may be of greater interest in view of new methods of computing entropy [W. S?omczyński, J. Kwapień, K. ?yczkowski, Entropy computing via integration over fractal measures, Chaos 10 (2000) 180-188].     

Absolutely continuous invariant measures for random maps with position dependent probabilities

A random map is discrete-time dynamical system in which one of a number of transformations is randomly selected and applied at each iteration of the process. Usually the map τ_k is chosen from a finite collection of maps with constant probability p_k. In this note we allow the p_k's to be functions of position. In this case, the random map cannot be considered to be a skew product. The main result provides a sufficient condition for the existence of an absolutely continuous invariant measure for position dependent random maps on [0,1]. Geometrical and topological properties of sets of absolutely continuous invariant measures, attainable by means of position dependent random maps, are studied theoretically and numerically.     

Thin spectra and singular continuous spectral measures for limit-periodic Jacobi matrices

This paper investigates the spectral properties of Jacobi matrices with limit-periodic coefficients. We show that generically the spectrum is a Cantor set of zero Lebesgue measure, and the spectral measures are purely singular continuous. For a dense set of limit-periodic Jacobi matrices, we show that the spectrum is a Cantor set of zero lower box counting dimension while still retaining the singular continuity of the spectral type. We also show how results of this nature can be established by fixing the off-diagonal coefficients and varying only the diagonal coefficients, and, in a more restricted version, by fixing the diagonal coefficients to be zero and varying only the off-diagonal coefficients. We apply these results to produce examples of weighted Laplacians on the multidimensional integer lattice having purely singular continuous spectral type and zero-dimensional spectrum.     

Continuous spectrum, point spectrum and residual spectrum of operator matrices

Let M_C denote a 2 × 2 upper triangular operator matrix of the form , which is acting on the sum of Banach spaces X⊕Y or Hilbert spaces H⊕K. In this paper, the sets and ?_C∈B(K,H)σ_r(M_C) are, respectively, characterized completely, where σ_c(·) denotes the continuous spectrum, σ_p(·) denotes the point spectrum and σ_r(·) denotes the residual spectrum. Moreover, some corresponding counterexamples are given.     

The spectrum of a random 3 × 3 matrix

In this paper, it is shown that the distribution of the spectrum of A + X gives (essentially) a complete set of orthogonal invariants for the (real nonsymmetric) matrix A.     

On the location of the discrete spectrum for complex Jacobi matrices

We study spectrum inclusion regions for complex Jacobi matrices that are compact perturbations of the discrete Laplacian. The condition sufficient for the lack of a discrete spectrum for such matrices is given.

     

Jacobi matrices with absolutely continuous spectrum

Let be a Jacobi matrix defined in as , where is a unilateral weighted shift with nonzero weights such that Define the seqences: If and , then has an absolutely continuous spectrum covering . Moreover, the asymptotics of the solution is also given.

     

Absolutely Continuous Spectrum of a Polyharmonic Operator with a Limit Periodic Potential in Dimension Two

We consider a polyharmonic operator H = (?Δ) ^l  + V(x) in dimension two with l ≥ 6, l being an integer, and a limit-periodic potential V(x). We prove that the spectrum contains a semiaxis of absolutely continuous spectrum.     

Discrete spectrum for complex perturbations of periodic Jacobi matrices

We study spectrum inclusion regions for complex Jacobi matrices, which are compact perturbations of real periodic Jacobi matrix. The condition sufficient for the lack of the discrete spectrum for such matrices is given.     

On the spectrum of a differential operator of high-order

In this paper, sufficient conditions for the spectrum of the operator of high order to be discrete and unbounded below are obtained.Translated from Matematicheskie Zametki, vol. 77, no. 2, 2005, pp. 188–193.Original Russian Text Copyright © 2005 by M. G. Gimadislamov.This revised version was published online in April 2005 with a corrected issue number.