Weyl–Titchmarsh M ‐Function Asymptotics for Matrix‐valued Schrödinger 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 ∈L loc 1 ( ( x 0 , ∞ ) ) m × m , where m ∈ N. More precisely, assume that for some N ∈ N and x 0 ∈R, Q ( N − 1 ) ∈ L 1( [ x 0 , c ) ) m × mfor all c > x 0 , and that x ⩾ x 0 is a right Lebesgue point of Q ( N –1) . In addition, denote by I m the m × m identity matrix and by C ɛ 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 I md 2 d x 2+ Q ( x ) inL 2 ( ( x 0 , ∞ ) ) m and a Dirichlet boundary condition at x = x 0 : M + ( z , x ) = | z | → ∞ , z ∈ C εi I m z 1 / 2 + ∑ k = 1 N m + , k ( x ) z − k / 2 + o ( | z | − N / 2 ) , where  N ∈ N . The expansion is uniform with respect to arg(z) for | z | → ∞ in C ɛ 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. Analogous results hold for matrix‐valued Schrödinger operators on the real line. 2000 Mathematics Subject Classification : 34E05, 34B20, 34L40, 34A55.