# Spectra of Jacobi operators via connection coefficient matrices

## TL;DR

This paper develops a method to compute the spectrum and spectral measure of Jacobi operators with compact perturbations using connection coefficient matrices, reducing the problem to polynomial root finding and establishing computability with error guarantees.

## Contribution

It introduces an explicit polynomial-based approach for spectral computation of perturbed Jacobi operators and proves the spectrum's computability within the Solvability Complexity Index framework.

## Key findings

- Spectrum computation reduces to polynomial root finding.
- Explicit formulas for spectral measure derived.
- Spectrum of compact perturbations is computable with error control.

## Abstract

We address the computational spectral theory of Jacobi operators that are compact perturbations of the free Jacobi operator via the asymptotic properties of a connection coefficient matrix. In particular, for finite-rank perturbation we show that the computation of the spectrum can be reduced to a polynomial root finding problem, from a polynomial that is derived explicitly from the entries of a connection coefficient matrix. A formula for the spectral measure of the operator is also derived explicitly from these entries. The analysis is extended to trace-class perturbations. We address issues of computability in the framework of the Solvability Complexity Index, proving that the spectrum of compact perturbations of the free Jacobi operator is computable in finite time with guaranteed error control in the Hausdorff metric on sets.