Jacobi matrices: continued fractions, approximation, spectrum

TL;DR

This paper investigates the spectral properties of self-adjoint operators represented by Jacobi matrices, using continued fractions to establish criteria for absolute continuity of the spectrum and analyzing spectral measures.

Contribution

It introduces new criteria for absolute continuity of the spectrum of Jacobi matrices based on continued fraction representations and operator approximation methods.

Findings

Criteria for absolute continuity of spectrum established

Conditions for spectral measure derivatives to be continuous derived

Approximation methods for spectral analysis developed

Abstract

In this work the spectral theory of self-adjoint operator $A$ represented by Jacobi matrix is considered. The approach is based on the continued fraction representation of the resolvent matrix element of $A$. Different criteria of absolute continuity of a spectrum are found. For the analysis of the absolutely continuous spectrum it is used an approximation of $A$ by a sequence of operators $A_n$ with absolutely continuous spectrum on a given interval $[a,b\,]$ which converges to $A$ in a strong sense on a dense set. In the case when $[a,b\,]\subseteq\sigma(A)$ it was found the sufficient condition of absolute continuity of the operator $A$ spectrum on $[a,b\,]$. This condition uses the notion of equi-absolute continuity. It is constructed the system of functions converging to the distribution function of the operator. In the case of the absolutely continuous spectrum, the system of continuous functions converging to the spectral weight of the operator on a given interval is also constructed and was analyzed. The conditions when the derivative of the distribution function of $A$ belongs to the class $C[a,b\,]$ are also obtained.