Approximation and asymptotics of eigenvalues of unbounded self-adjoint Jacobi matrices acting in 12 by the use of finite submatrices

摘要

We consider the problem of approximation of eigenvalues of a self-adjoint operator 1 defined by a Jacobi matrixin the Hilbert space 12(N) by eigenvalues of principal finite submatrices of an infinite Jacobi matrix that definesthis operator. We assume the operator 1 is bounded from below with compact resolvent. In our research weestimate the asymptotics (with noo) of the joint error of approximation for the eigenvalues, numbered from1 to N, of ] by the eigenvalues of the finite submatrix ∫nof ordern x n,whereN =max{k E N : < rn}and r E (0,1) is arbitrary chosen. We apply this result to obtain an asymptotics for the eigenvalues of 1. Themethod applied in this research is based on Volkmer's results included in [23].