# A point interaction for the discrete Schr\"odinger operator and generalized Chebyshev polynomials

## TL;DR

This paper investigates a special class of Jacobi matrices representing point interactions in discrete Schrödinger operators, deriving explicit spectral characteristics and introducing a new class of orthogonal polynomials that generalize Chebyshev polynomials.

## Contribution

It provides explicit formulas for spectral measures and resolvents of these matrices and introduces a novel class of orthogonal polynomials extending Chebyshev polynomials.

## Key findings

- Explicit spectral measure formulas derived
- New class of orthogonal polynomials introduced
- Spectral analysis of point interactions achieved

## Abstract

We consider semi-infinite Jacobi matrices corresponding to a point interaction for the discrete Schr\"odinger operator. Our goal is to find explicit expressions for the spectral measure, the resolvent and other spectral characteristics of such Jacobi matrices. It turns out that their spectral analysis leads to a new class of orthogonal polynomials generalizing the classical Chebyshev polynomials.