On 2D discrete Schr\"odinger operators associated with multiple orthogonal polynomials

TL;DR

This paper introduces a class of 2D difference operators linked to multiple orthogonal polynomials, generalizing classical orthogonal polynomial connections to lattice operators and constructing explicit 2D discrete Schrödinger operators.

Contribution

It develops a novel framework connecting 2D difference operators with multiple orthogonal polynomials and constructs explicit 2D discrete Schrödinger operators from this scheme.

Findings

Operators' eigenvectors are multiple orthogonal polynomials

Generalization of Jacobi matrix-orthogonal polynomial connection to 2D

Explicit examples of 2D discrete Schrödinger operators

Abstract

A class of cross-shaped difference operators on a two dimensional lattice is introduced. The main feature of the operators in this class is that their formal eigenvectors consist of multiple orthogonal polynomials. In other words, this scheme generalizes the classical connection between Jacobi matrices and orthogonal polynomials to the case of operators on lattices. Furthermore we also show how to obtain 2D discrete Schr\"odinger operators out of this construction and give a number of explicit examples based on known families of multiple orthogonal polynomials.