Weyl-Titchmarsh M -Function Asymptotics for Matrix-valued Schrödinger Operators

Abstract

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 x0∈R, Q ( N − 1 ) ∈ L 1 ( [ x 0 , 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 I m d 2 d x 2 + Q ( x ) in L 2 ( ( x 0 , ∞ ) ) m and a Dirichlet boundary condition at x=x0: M + ( z , x ) = | z | → ∞ , z ∈ C ε i 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 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. Analogous results hold for matrix-valued Schrödinger operators on the real line. 2000 Mathematics Subject Classification: 34E05, 34B20, 34L40, 34A55.