Jost asymptotics for matrix orthogonal polynomials on the real line

TL;DR

This paper extends classical Jost asymptotics to matrix-valued orthogonal polynomials on the real line, providing conditions for Jost functions and characterizing matrix-valued Weyl-Titchmarsh m-functions for block Jacobi matrices.

Contribution

It introduces matrix-valued Jost asymptotics under L1 conditions and characterizes the associated Weyl-Titchmarsh functions, generalizing previous scalar results.

Findings

Established matrix-valued Jost asymptotics for block Jacobi matrices.

Provided necessary and sufficient conditions for Jost functions.

Fully characterized matrix-valued Weyl-Titchmarsh m-functions.

Abstract

We obtain matrix-valued Jost asymptotics for block Jacobi matrices under an L1-type condition on Jacobi parameters, and give a necessary and sufficient condition for an analytic matrix-valued function to be the Jost function of a block Jacobi matrix with exponentially converging parameters. This establishes the matrix-valued analogue of Damanik-Simon-II paper [6]. The above results allow us to fully characterize the matrix-valued Weyl-Titchmarsh m-functions of block Jacobi matrices with exponentially converging parameters.