# The Inverse Spectral Problem for Jacobi-Type Pencils

## TL;DR

This paper investigates the inverse spectral problem for Jacobi-type pencils, which are matrix pencils involving Jacobi matrices and five-diagonal matrices, providing explicit spectral functions for certain polynomial perturbations.

## Contribution

It introduces a framework for solving the inverse spectral problem for Jacobi-type pencils and derives explicit spectral functions for specific polynomial perturbations.

## Key findings

- Explicit spectral functions for special polynomial perturbations
- Framework for inverse spectral problem for Jacobi-type pencils
- Analysis of spectral properties of five-diagonal matrices

## Abstract

In this paper we study the inverse spectral problem for Jacobi-type pencils. By a Jacobi-type pencil we mean the following pencil $J_5 - \lambda J_3$, where $J_3$ is a Jacobi matrix and $J_5$ is a semi-infinite real symmetric five-diagonal matrix with positive numbers on the second subdiagonal. In the case of a special perturbation of orthogonal polynomials on a finite interval the corresponding spectral function takes an explicit form.