On direct and inverse spectral problems for a class of selfadjoint difference operators on the graph–tree

Abstract

We consider selfadjoint operators on the graph-tree which are constructed by means of the difference equations, connecting nearest neighbors in the lattice of the Multiple Orthogonal Polynomials (MOPs). This construction generalizes the Jacobi matrices formed by the coefficients of the recurrence relations for Orthogonal Polynomials. We discuss the connections between the spectral measure of this operator, the vector-measure of multiple orthogonality and the limits of the nearest neighbors recurrence coefficients along the rays in the lattice of MOPs. The solvability of the direct spectral problem: existence of the operator coefficients ray sequences whose limits determine the support of the spectral measure are proven. The solution of the inverse problem: find the corresponding limit of the coefficients starting from the spectral measure support, are given.We consider selfadjoint operators on the graph-tree which are constructed by means of the difference equations, connecting nearest neighbors in the lattice of the Multiple Orthogonal Polynomials (MOPs). This construction generalizes the Jacobi matrices formed by the coefficients of the recurrence relations for Orthogonal Polynomials. We discuss the connections between the spectral measure of this operator, the vector-measure of multiple orthogonality and the limits of the nearest neighbors recurrence coefficients along the rays in the lattice of MOPs. The solvability of the direct spectral problem: existence of the operator coefficients ray sequences whose limits determine the support of the spectral measure are proven. The solution of the inverse problem: find the corresponding limit of the coefficients starting from the spectral measure support, are given.