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 Schrodinger operator defined on the interval $ [0,b) $ [0,b), where $ 0<b\leq \infty $ 0<b <=infinity. Let $ H_j=-\frac {d<^>2}{dx<^>2}I_m+Q_j $ Hj=-d2dx2Im+Qj, j=1,2, be two self-adjoint Schrodinger operators in $ L<^>2((0,b))<^>{m\times m} $ L2((0,b))mxm and $ Q_1=Q_2 $ Q1=Q2 a.e. on the interval $ [0,a] $ [0,a], where $ a\in (0,b) $ a is an element of(0,b). It is assumed that the potentials $ Q_1 $ Q1 and $ Q_2 $ Q2 are sufficiently smooth in the right neighborhood of the point a, where the right-hand derivatives of $ Q_1=Q_2 $ Q1=Q2 at a coincide up to a certain order. Let $ M_j(z) $ Mj(z) be the Weyl-Titchmarsh functions of $ H_j=-\frac {{\rm d}<^>2}{{\rm d}x<^>2}I_m+Q_j $ Hj=-d2dx2Im+Qj, j=1,2. As a specific application of this expression, we establish a high-energy asymptotic for the difference between $ M_1(z) $ M1(z) and $ M_2(z) $ M2(z). Besides, new proofs are given for the local Borg-Marchenko uniqueness theorem and the high-energy asymptotics of the Weyl-Titchmarsh functions.