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.

Multiple Askey-Wilson polynomials and related basic hypergeometric multiple orthogonal polynomials

“…Often one can take the orthogonality measures for classical orthogonal polynomials and by allowing r different parameters one gets r measures with respect to which one can look for the corresponding multiple orthogonal polynomials, see, e.g., [2,7,30]. Some of these 'classical' multiple orthogonal polynomials play an important role in applications, e.g., multiple Hermite polynomials and multiple Laguerre polynomials are used in the analysis of random matrices [10,11,18] or special determinantal processes [19], multiple Jacobi polynomials and multiple little q-Jacobi polynomials are used in irrationality proofs [26,27,28], multiple Charlier and multiple Meixner polynomials are used to describe non-Hermitian oscillator Hamiltonians [21,22,23], and in general multiple orthogonal polynomials they are useful in the analysis of multidimensional Schrödinger equations and the multidimensional Toda lattice [3,4].…”

Multidimensional Toda Lattices: Continuous and Discrete Time

“…Here we follow the concept of discrete integrability given in [20]. It is now clear that the consistency of (3.40) gives (3.39), which is in fact a discrete integrable system [13,14]. Namely, in [14] and [55] it is shown that the discrete zero curvature condition (3.39) is equivalent to the nonlinear system of difference equations (1.7) for the coefficients of the recurrence relations (3.38 (a n+1,i,1 + a n+1,i,2 ) − (a n,i+1,1 + a n,i+1,2 )…”

Hypercomplex Fock states for discrete electromagnetic Schrödinger operators: A Bayesian probability perspective

“…Discrete electromagnetic Schrödinger operators correspond to a subclass of (doubly) Jacobi operators. They are ubiquitous in several fields of mathematics, physics and beyond, as is witnessed by the papers [16,18,35,38,31,8,6,37,24,1] and monograph [36]. Here, the factorization method is the cornerstone in the study of the quasi-exact solvability of such kind of operators since it avoids non-perturbative arguments that appear under the discretization of its continuum counterpart, the quantum harmonic oscillator − 1 2m ∆ + V (x) with mass m and potential V (x) (cf.…”