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 [ 0 , b ) , where 0 < b ≤ ∞ . Let H j = − d 2 d x 2 I m + Q j , j=1,2, be two self-adjoint Schrödinger operators in L 2 ( ( 0 , b ) ) m × m and Q 1 = Q 2 a.e. on the interval [ 0 , a ] , where a ∈ ( 0 , b ) . It is assumed that the potentials Q 1 and Q 2 are sufficiently smooth in the right neighborhood of the point a, where the right-hand derivatives of Q 1 = Q 2 at a coincide up to a certain order. Let M j ( z ) be the Weyl-Titchmarsh functions of H j = − d 2 d x 2 I m + Q j , j=1,2. As a specific application of this expression, we establish a high-energy asymptotic for the difference between M 1 ( z ) and M 2 ( z ) . Besides, new proofs are given for the local Borg-Marchenko uniqueness theorem and the high-energy asymptotics of the Weyl-Titchmarsh functions.