Weyl-titchmarsh M -function Asymptotics for Matrix-valued Schr¨odinger Operators

We explicitly determine the high‐energy asymptotics for Weyl–Titchmarsh matrices corresponding to matrix‐valued Schrödinger operators associated with general self‐adjoint m × m matrix potentials Q∈Lloc1((x0,∞))m×m , where m ∈ N. More precisely, assume that for some N ∈ N and x0∈R, Q(N−1)∈L1([x0,c))m×m for all c>x0, and that x⩾ x0 is a right Lebesgue point of Q(N–1). In addition, denote by Im the m×m identity matrix and by Cɛ the open sector in thecomplex plane with vertex at zero, symmetry axis along the positive imaginary axis, and opening angle ɛ, with 0 < ε < ½π. Then we prove the following asymptotic expansion for any point M+(z,x) of the unique limit point or a point of the limit disk associated with the differential expression Imd2dx2+Q(x) in L2((x0,∞))m and a Dirichlet boundary condition at x=x0: M+(z,x)=|z|→∞,z∈CεiImz1/2+∑k=1Nm+,k(x)z−k/2+o(|z|−N/2),where N∈N. The expansion is uniform with respect to arg(z) for |z| → ∞ in Cɛ and uniform in x as long as x varies in compact subsets of R intersected with the right Lebesgue set of Q(N–1). Moreover, the m × m expansion coefficients m+,k(x) can be computed recursively.