Asymptotics of Solutions of Some Integral Equations Connected with Differential Systems with a Singularity

Our studies concern some aspects of scattering theory of the singular differential systems y' - x(-1)Ay - q(x)y = rho By, x > 0 with n x n matrices A, B, q(x), x is an element of (0, infinity), where A, B are constant and rho is a spectral parameter. We concentrate on investigation of certain Volterra integral equations with respect to tensor-valued functions. The solutions of these integral equations play a central role in construction of the so-called Weyl-type solutions for the original differential system. Actually, the integral equations provide a method for investigation of the analytical and asymptotical properties of the Weyl-type solutions while the classical methods fail because of the presence of the singularity. In the paper, we consider the important special case when q is smooth and q(0) = 0 and obtain the classical-type asymptotical expansions for the solutions of the con- sidered integral equations as rho -> infinity with o (rho(-1)) rate remainder estimate. The result allows one to obtain analogous asymptotics for the Weyl-type solutions that play in turn an important role in the inverse scattering theory.