Discrete Schrödinger Operator on a Tree, Angelesco Potentials, and Their Perturbations

Abstract

We consider a class of discrete Schrodinger operators on an infinite homogeneous rooted tree. Potentials for these operators are given by the coefficients of recurrence relations satisfied on a multidimensional lattice by multiple orthogonal polynomials. For operators on a binary tree with potentials generated by multiple orthogonal polynomials with respect to systems of measures supported on disjoint intervals (Angelesco systems) and for compact perturbations of such operators, we show that the essential spectrum is equal to the union of the intervals supporting the orthogonality measures.