Weyl-Titchmarsh \(M\)-function asymptotics for matrix-valued Schrödinger operators (Q2766405)

The matrix-valued Schrödinger operator \(H_+=-I_m\frac{d^2}{dx^2}+Q\) is considered in \(L^2((x_0,\infty))^m\), where \(I_m\) is the identity matrix in \(\mathbb{C}^m\), \(m\in\mathbb{N}\), \(Q=Q^*\in L^1([x_0,c])^{m\times m}\) for all \(c>x_0\), and certain boundary conditions are satisfied. The authors study the Weyl-Titchmarsh matrix \(M_+(z,x_0)\) associated with the above differential expression. More precisely, a systematic higher-order high-energy asymptotic expansion of \(M_+(z,x_0)\) as \(z\to\infty\) is developed, combining Atkinson's (apparently unpublished) result (his approach is described in the paper) with matrix-valued extensions of some methods based on an associated Riccati equation.