© 2000 by London Mathematical Society
Weyl–Titchmarsh M-Function Asymptotics for Matrix-valued Schrödinger Operators
Department of Mathematics and Statistics, University of Missouri–Rolla Rolla, MO 65409, USA; sclark{at}umr.edu; http://www.umr.edu/clark
Department of Mathematics, University of Missouri Columbia, MO 65211, USA; fritz{at}math.missouri.edu; http://www.math.missouri.edu/people/fgesztesy.html
Received 18 May 1999. Revision received 28 March 2000.
We explicitly determine the high-energy asymptotics for Weyl–Titchmarsh matrices corresponding to matrix-valued Schrödinger operators associated with general self-adjoint m x m matrix potentials 
, where m 
N. More precisely, assume that for some N N and x0R, 
for all c>x0, and that x
x0 is a right Lebesgue point of Q(N–1). In addition, denote by Im the mxm identity matrix and by C
the open sector in thecomplex plane with vertex at zero, symmetry axis along the positive imaginary axis, and opening angle , with 0 < < 
. 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 
in 
and a Dirichlet boundary condition at x=x0:
The expansion is uniform with respect to arg(z) for |z| 
in C 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 x 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.
Key Words: Weyl–Titchmarsh functions • Schrödinger operators • high-energy asymptotics