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

ABSTRACT 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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    • "As it is pointed out in [10], none of the conditions (B 1 ) or (B 2 ) is weaker than the other. Indeed, we will see that this also holds for (B 1 )–(B 4 ), and in general for the conditions obtained from any bound F m,N , which become complementary (see §5). "
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    ABSTRACT: 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 conditions given in both cases improve and generalize previously known results. It turns out that, not only self-adjointness helps to study limit points of eigenvalues of truncated operators, but the analysis of such limit points is a key help to prove self-adjointness. Several examples show the advantages of these new results compared with previous ones. Besides, an application to the theory of continued fractions is pointed out.
    Journal of Mathematical Analysis and Applications 02/2014; 420(1). DOI:10.1016/j.jmaa.2014.05.077 · 1.12 Impact Factor
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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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