On 2D discrete Schrödinger operators associated with multiple orthogonal polynomials

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 Schrödinger operators out of this construction and give a number of explicit examples based on known families of multiple orthogonal polynomials.

15 pages

References in corpus (5)

• The nearest neighbor recurrence coefficients for multiple orthogonal polynomials
• The essential spectrum of Schrödinger operators on lattices
• Asymptotic analysis of random matrices with external source and a family of algebraic curves
• Multiple Meixner polynomials and non-Hermitian oscillator Hamiltonians
• Two-dimensional algebro-geometric difference operators