The local Borg-Marchenko uniqueness theorem for matrix-valued Schrödinger operators with locally smooth at the right endpoint potentials

Abstract

We present a new expression for the Weyl-Titchmarsh matrix-valued function of a self-adjoint matrix-valued Schrödinger operator defined on the interval , where . Let , j=1,2, be two self-adjoint Schrödinger operators in and a.e. on the interval , where . It is assumed that the potentials and are sufficiently smooth in the right neighborhood of the point a, where the right-hand derivatives of at a coincide up to a certain order. Let be the Weyl-Titchmarsh functions of , j=1,2. As a specific application of this expression, we establish a high-energy asymptotic for the difference between and . Besides, new proofs are given for the local Borg-Marchenko uniqueness theorem and the high-energy asymptotics of the Weyl-Titchmarsh functions.