Asymptotics of solutions of some integral equations connected with differential systems with a singularity

M. Yu. Ignatiev

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 20 (2020) no. 1, pp. 17-28

Résumé

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 $n\times n$ matrices $A,B, q(x), x\in(0,\infty)$, 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 considered integral equations as $\rho\to\infty$ with $o\left(\rho^{-1}\right)$ 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.