Approximation of eigenvalues of some unbounded self-adjoint discrete Jacobi matrices by eigenvaules of finite submatrices

• Autorzy: Malejki M.
• Języki publikacji: EN

Abstrakty

EN

We investigate the problem of approximation of eigenvalues of some self-adjoint operator in the Hilbert space l2(N) by eigenvalues of suitably chosen principal finite submatrices of an infinite Jacobi matrix that defines the operator considered. We assume the Jacobi operator is bounded from below with compact resolvent. In our research we estimate the asymptotics (with n → ∞) of the joint error of approximation for the first n eigenvalues and eigenvectors of the operator by the eigenvalues and eigenvectors of the finite submatrix of order n x n. The method applied in our research is based on the Rayleigh-Ritz method and Volkmer's results included in [7]. We extend the method to cover a class of infinite symmetric Jacobi matrices with three diagonals satisfying some polynomial growth estimates.

Słowa kluczowe

EN

self-adjoint unbounded Jacobi matrix; asymptotics; point spectrum; tridiagonal matrix; eigenvalue

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2007
• Tom: Vol. 27, no. 1
• Strony: 37--49
• Opis fizyczny: Bibliogr. 7 poz.

Twórcy

autor

Malejki M.

• AGH University of Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Cracow, Poland,

Bibliografia

• [1] J. Janas, M. Malejki, Alternative approaches to asymptotic behaviour of eigenvalues of some unbounded Jacobi matrices, J. Comp. and Appl. Math. 200 (2007), 342-356.
• [2] J. Janas, S. Naboko, Multithreshold Spectral Phase Transitions for a Class of Jacobi Matrices, Operator Theory: Adv. Appl. 124 (2001), 267-285.
• [3] J. Janas, S. Naboko, Infinite Jacobi matrices with unbounded entries: asymptotics of eigenvalues and the transformation operator approach, SIAM J. Math. Anal. 36 (2004), 2, 643-658.
• [4] G.V. Rozenbljum, Near-similarity of operators and the spectral asymptotic behaviour of pseudodifferential operators on the circle, Trudy Maskov. Mat. Obshch. 36 (1978), 59-84 (in Russian).
• [5] H. Weyl, Das asymtotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Math. Ann. 71 (1912), 441-479 (in German).
• [6] J.H. Wilkinson, Rigorous error bounds for coputed eigensystems, Comput. J. 4 (1961), 230-241.
• [7] H. Volkmer, Error Estimates for Rayleigh-Ritz Approximations of Eigenvalues and Eigenfunctions of the Mathieu and Spheroidal Wave Equation, Constr. Approx. 20 (2004), 39-54.