Limit points of eigenvalues of truncated unbounded tridiagonal operators

Central European Journal of Mathematics (Impact Factor: 0.58). 05/2007; 5(2):335-344. DOI: 10.2478/s11533-007-0009-1


Let T be a self-adjoint tridiagonal operator in a Hilbert space H with the orthonormal basis {e



, σ(T) be the spectrum of T and Λ(T) be the set of all the limit points of eigenvalues of the truncated operator T

. We give sufficient conditions such that the spectrum of T is discrete and σ(T) = Λ(T) and we connect this problem with an old problem in analysis.

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    • "In addition, as an important step of the proof, one makes use of an explicit formula for the Green function associated with J. Apart of the localization of the spectrum we address too the question of approximation of the spectrum by spectra of truncated finite-dimensional Jacobi matrices. For bounded Hermitian Jacobi operators the problem has been studied, for example in [11] [3] [13]. We are aware of just a few papers, however, bringing some results in this respect also about unbounded Jacobi operators [14] [12] [18]. "
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    ABSTRACT: We introduce a class of Jacobi operators with discrete spectra which is characterized by a simple convergence condition. With any operator J from this class we associate a characteristic function as an analytic function on a suitable domain, and show that its zero set actually coincides with the set of eigenvalues of J in that domain. Further we derive sufficient conditions under which the spectrum of J is approximated by spectra of truncated finite-dimensional Jacobi matrices. As an application we construct several examples of Jacobi matrices for which the characteristic function can be expressed in terms of special functions. In more detail we study the example where the diagonal sequence of J is linear while the neighboring parallels to the diagonal are constant.
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    • "So, asymptotic and approximate approaches have to be applied (see, e.g., [3–5,12,13,17,21] and [22]). Projective methods, that use finite submatrices to investigate spectral properties of operators given by infinite Jacobi matrices are applied successfully (see [1] [2] [10] [15] [16] [21]). "
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    ABSTRACT: The research included in the paper concerns a class of symmetric block Jacobi matrices. The problem of the approximation of eigenvalues for a class of a self-adjoint unbounded operators is considered. We estimate the joint error of approximation for the eigenvalues, numbered from 1 to $N$, for a Jacobi matrix $J$ by the eigenvalues of the finite submatrix $J_n$ of order $pn \times pn$, where $N = \max \{k \in N : k \leq rpn\}$ and $r \in (0,1)$ is suitably chosen. We apply this result to obtain the asymptotics of the eigenvalues of $J$ in the case $p=3$.
    Opuscula Mathematica 01/2010; 30(3). DOI:10.7494/OpMath.2010.30.3.311
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    • "In the theory of orthogonal polynomials it is known that if ¡ 1 4 then the spectrum of the tridiagonal operator T is discrete, consisting of a denumerable set of values n such that lim| n |=∞, n → ∞ [1] [3]. Also, from well-known results of the theory of orthogonal polynomials [1], it follows [2] that if T is bounded from below, i.e., (Tx; x)¿x 2 , ∈ R, x in the deÿnition domain of T and if = 1 4 then the spectrum of T is purely continuous and consists of the interval [; ∞). "
    Journal of Computational and Applied Mathematics 08/2001; 133(1):688-689. DOI:10.1016/S0377-0427(00)00721-4 · 1.27 Impact Factor
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