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

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) $ [ 0 , b) , where $ 0 0 < b ≤ ∞. Let $ H_j=-\frac {d^2}{dx^2}I_m+Q_j $ 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\times m} $ L 2 ((0 , b)) m × m and $ Q_1=Q_2 $ Q 1 = Q 2 a.e. on the interval $ [0,a] $ [ 0 , a ] , where $ a\in (0,b) $ a ∈ (0 , b). It is assumed that the potentials $ Q_1 $ Q 1 and $ Q_2 $ Q 2 are sufficiently smooth in the right neighborhood of the point a, where the right-hand derivatives of $ Q_1=Q_2 $ Q 1 = Q 2 at a coincide up to a certain order. Let $ M_j(z) $ M j (z) be the Weyl-Titchmarsh functions of $ H_j=-\frac {{\rm d}^2}{{\rm d}x^2}I_m+Q_j $ 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) $ M 1 (z) and $ M_2(z) $ 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. [ABSTRACT FROM AUTHOR]