相关论文: Finite differrence operators with a finite--band s…

We offer a detailed treatment of spectral and Weyl-Titchmarsh-Kodaira theory for all self-adjoint Jacobi operator realizations of the differential expression \begin{align*} \tau_{\alpha,\beta} = - (1-x)^{-\alpha} (1+x)^{-\beta}(d/dx)…

In this paper, we present an approach for explicitly constructing quasi-periodic Schr\"odinger operators with Cantor spectrum with $C^k$ potential. Additionally, we provide polynomial asymptotics on the size of spectral gaps.

We give necessary and sufficient conditions for real sequences to be the spectra of selfadjoint extensions of an entire operator whose domain may be non-dense. For this spectral characterization we use de Branges space techniques and a…

Borg-type uniqueness theorems for matrix-valued Jacobi operators H and supersymmetric Dirac difference operators D are proved. More precisely, assuming reflectionless matrix coefficients A, B in the self-adjoint Jacobi operator H=AS^+ +…

We introduce the periodic Airy-Schr\"odinger operator and we study its band spectrum. This is an example of an explicitly solvable model with a periodic potential which is not differentiable at its minima and maxima. We define a…

The goal of this note is to study the spectrum of a self-adjoint convolution operator in $L^2(\mathbb R^d)$ with an integrable kernel that is perturbed by an essentially bounded real-valued potential tending to zero at infinity. We show…

We consider the periodic Jacobi operator $J$ with finitely supported perturbations on the half-lattice. We describe all eigenvalues and resonances of $J$ and give their properties. We solve the inverse resonance problem: we prove that the…

This paper addresses two different but related questions regarding an unbounded symmetric tridiagonal operator: its self-adjointness and the approximation of its spectrum by the eigenvalues of its finite truncations. The sufficient…

We survey results that have been obtained for self-adjoint operators, and especially Schr\"odinger operators, associated with mathematical models of quasicrystals. After presenting general results that hold in arbitrary dimensions, we focus…

We define a class of discrete operators acting on infinite, finite or periodic sequences mimicking the standard properties of pseudo-differential operators. In particular we can define the notion of order and regularity, and we recover the…

We study discrete Schroedinger operators with analytic potentials. In particular, we are interested in the connection between the absolutely continuous spectrum in the almost periodic case and the spectra in the periodic case. We prove a…

We consider the problem of embedding eigenvalues into the essential spectrum of periodic Jacobi operators, using an oscillating, decreasing potential. To do this we employ a geometric method, previously used to embed eigenvalues into the…

We present a new oscillation criterion to determine whether the number of eigenvalues below the essential spectrum of a given Jacobi operator is finite or not. As an application we show that Kenser's criterion for Jacobi operators follows…

Several examples of Jacobi matrices with an explicitly solvable spectral problem are worked out in detail. In all discussed cases the spectrum is discrete and coincides with the set of zeros of a special function. Moreover, the components…

Spectral operators of matrices proposed recently in [C. Ding, D.F. Sun, J. Sun, and K.C. Toh, Math. Program. {\bf 168}, 509--531 (2018)] are a class of matrix valued functions, which map matrices to matrices by applying a vector-to-vector…

In this paper, we briefly explain the spectral expansion problem for differential operators defined on the entire real line, generated by a differential expression with periodic, complex-valued coefficients.

We develop singular Weyl-Titchmarsh-Kodaira theory for Jacobi operators. In particular, we establish existence of a spectral transformation as well as local Borg-Marchenko and Hochstadt-Liebermann type uniqueness results.

We study spectral properties of second order elliptic operators with periodic coefficients in dimension two. These operators act in periodic simply-connected waveguides, with either Dirichlet, or Neumann, or the third boundary condition.…

We present a technique for reconstructing a semi-infinite Jacobi operator in the limit circle case from the spectra of two different self-adjoint extensions. Moreover, we give necessary and sufficient conditions for two real sequences to be…

A short introduction to the use of the spectral theorem for self-adjoint operators in the theory of special functions is given. As the first example, the spectral theorem is applied to Jacobi operators, i.e. tridiagonal operators, on…