Self-adjointness of unbounded tridiagonal operators and spectra of their finite truncations

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.Keywords: unbounded symmetric operators, Jacobi matrices, self-adjointness, spectrum of an operator, limit points of eigenvalues, zeros of orthogonal polynomials, Jacobi continued fractions. 2000 MSC: 47B25, 47B36

On unbounded commuting Jacobi operators and some related issues

“…Thus, for di erent values of p, we have a number of su cient conditions for the operator L to be self-adjoint. As noted in [14], some of these conditions coincide with the previously known results on the Jacobi operators (see also [11], where various su cient conditions are studied). For example, for p = we nd using (2) that…”

On the Deficiency Indices of Block Jacobi Matrices Related to Dirac Operators with Point Interactions

“…This topic has attracted substantial attention, in particular during the last two decades (see for instance, [9,10,12,13,15,17,23,24,25,32,33,34,35,36,37,38,39,40,46]). We especially mention recent publications [47] (p = 1) and [10], [15] (p ≥ 1) where new different conditions for block Jacobi matrices to be selfadjoint were found. Besides, in [15], [12], [13] several discreteness conditions for these matrices were established.…”