Cite this article as:
Ignatyev M. Y. Asymptotics of Solutions of Some Integral Equations Connected with Differential Systems with a Singularity. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2020, vol. 20, iss. 1, pp. 17-28. DOI: https://doi.org/10.18500/1816-9791-2020-20-1-17-28
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 −1Ay − q(x)y = ρBy,
x 0 with n × n matrices A, B, q(x), x ∈ (0, ∞), where A, B are constant and ρ 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 ρ → ∞ with o $ ρ −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.
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