Approximation and asymptotics of eigenvalues of unbounded self-adjoint Jacobi matrices acting in l 2 by the use of finite submatrices

Open Mathematics (2010)

• Volume: 8, Issue: 1, page 114-128
• ISSN: 2391-5455

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Abstract

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We consider the problem of approximation of eigenvalues of a self-adjoint operator J defined by a Jacobi matrix in the Hilbert space l 2(ℕ) by eigenvalues of principal finite submatrices of an infinite Jacobi matrix that defines this operator. We assume the operator J is bounded from below with compact resolvent. In our research we estimate the asymptotics (with n → ∞) of the joint error of approximation for the eigenvalues, numbered from 1 to N; of J by the eigenvalues of the finite submatrix J n of order n × n; where N = max{k ∈ ℕ: k ≤ rn} and r ∈ (0; 1) is arbitrary chosen. We apply this result to obtain an asymptotics for the eigenvalues of J. The method applied in this research is based on Volkmer’s results included in [23].

How to cite

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Maria Malejki. "Approximation and asymptotics of eigenvalues of unbounded self-adjoint Jacobi matrices acting in l 2 by the use of finite submatrices." Open Mathematics 8.1 (2010): 114-128. <http://eudml.org/doc/269506>.

@article{MariaMalejki2010,
abstract = {We consider the problem of approximation of eigenvalues of a self-adjoint operator J defined by a Jacobi matrix in the Hilbert space l 2(ℕ) by eigenvalues of principal finite submatrices of an infinite Jacobi matrix that defines this operator. We assume the operator J is bounded from below with compact resolvent. In our research we estimate the asymptotics (with n → ∞) of the joint error of approximation for the eigenvalues, numbered from 1 to N; of J by the eigenvalues of the finite submatrix J n of order n × n; where N = max\{k ∈ ℕ: k ≤ rn\} and r ∈ (0; 1) is arbitrary chosen. We apply this result to obtain an asymptotics for the eigenvalues of J. The method applied in this research is based on Volkmer’s results included in [23].},
author = {Maria Malejki},
journal = {Open Mathematics},
keywords = {Self-adjoint unbounded Jacobi matrix; Asymptotics; Point spectrum; Tridiagonal matrix; Eigenvalue; selfadjoint unbounded Jacobi matrix; asymptotics; point spectrum; tridiagonal matrix; eigenvalues},
language = {eng},
number = {1},
pages = {114-128},
title = {Approximation and asymptotics of eigenvalues of unbounded self-adjoint Jacobi matrices acting in l 2 by the use of finite submatrices},
url = {http://eudml.org/doc/269506},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Maria Malejki
TI - Approximation and asymptotics of eigenvalues of unbounded self-adjoint Jacobi matrices acting in l 2 by the use of finite submatrices
JO - Open Mathematics
PY - 2010
VL - 8
IS - 1
SP - 114
EP - 128
AB - We consider the problem of approximation of eigenvalues of a self-adjoint operator J defined by a Jacobi matrix in the Hilbert space l 2(ℕ) by eigenvalues of principal finite submatrices of an infinite Jacobi matrix that defines this operator. We assume the operator J is bounded from below with compact resolvent. In our research we estimate the asymptotics (with n → ∞) of the joint error of approximation for the eigenvalues, numbered from 1 to N; of J by the eigenvalues of the finite submatrix J n of order n × n; where N = max{k ∈ ℕ: k ≤ rn} and r ∈ (0; 1) is arbitrary chosen. We apply this result to obtain an asymptotics for the eigenvalues of J. The method applied in this research is based on Volkmer’s results included in [23].
LA - eng
KW - Self-adjoint unbounded Jacobi matrix; Asymptotics; Point spectrum; Tridiagonal matrix; Eigenvalue; selfadjoint unbounded Jacobi matrix; asymptotics; point spectrum; tridiagonal matrix; eigenvalues
UR - http://eudml.org/doc/269506
ER -

References

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