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Bounds on the exponential decay of generalized eigenfunctions of bounded and unbounded selfadjoint Jacobi matrices are established. Two cases are considered separately: (i) the case in which the spectral parameter lies in a general gap of the spectrum of the Jacobi matrix and (ii) the case of a lower semi-bounded Jacobi matrix with values of the spectral parameter below the spectrum. It is demonstrated by examples that both results are sharp. We apply these results to obtain a many barriers-type criterion for the existence of square-summable generalized eigenfunctions of an unbounded Jacobi matrix at almost every value of the spectral parameter in suitable open sets. As an application, we provide examples of unbounded Jacobi matrices with a spectral mobility edge.
We study spectrum inclusion regions for complex Jacobi matrices which are compact perturbations of the discrete laplacian. The condition sufficient for the lack of discrete spectrum for such matrices is given.
We consider a class of Jacobi matrices with unbounded entries in the so called critical (double root, Jordan box) case. We prove a formula for the spectral density of the matrix which relates its spectral density to the asymptotics of orthogonal polynomials associated with the matrix.
We continue studying the connection between Jacobi matrices defined on a tree and multiple orthogonal polynomials (MOPs) that was discovered previously by the authors. In this paper, we consider Angelesco systems formed by two analytic weights and ob
For a nonnegative self-adjoint operator $A_0$ acting on a Hilbert space $mathfrak{H}$ singular perturbations of the form $A_0+V, V=sum_{1}^{n}{b}_{ij}<psi_j,cdot>psi_i$ are studied under some additional requirements of symmetry imposed on the initia
We show that for a Jacobi operator with coefficients whose (j+1)th moments are summable the jth derivative of the scattering matrix is in the Wiener algebra of functions with summable Fourier coefficients. We use this result to improve the known disp