Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

For citation:

Ignatyev M. Y. Asymptotics of Solutions of Some Integral Equations Connected with Differential Systems with a Singularity. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2020, vol. 20, iss. 1, pp. 17-28. DOI: 10.18500/1816-9791-2020-20-1-17-28, EDN: QEZGRI

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
Full text:
(downloads: 505)
Article type: 

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

Ignatyev M. Yu., Saratov State University

Our studies concern some aspects of scattering theory of the singular differential systems y′ − x−1Ayq(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−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.

  1. Brunnhuber R., Kostenko A., Teschl G. Singular Weyl – Titchmarsh – Kodaira theory for one-dimensional Dirac operators. Monatshefte fur Mathematik ¨ , 2014, vol. 174, pp. 515– 547. DOI: https://doi.org/10.1007/s00605-013-0563-5
  2. Albeverio S., Hryniv R., Mykytyuk Ya. Reconstruction of radial Dirac operators. J. Math. Phys., 2007, vol. 48, iss. 4, 043501, 14 p. DOI: https://doi.org/10.1063/1.2709847
  3. Albeverio S., Hryniv R., Mykytyuk Ya. Reconstruction of radial Dirac and Schr´’odinger operators from two spectra. J. Math. Anal. Appl., 2008, vol. 339, iss. 1, pp. 45–57. DOI: https://doi.org/10.1016/j.jmaa.2007.06.034
  4. Serier F. Inverse Problems Inverse spectral problem for singular Ablowitz–Kaup–Newell–Segur operators on [0, 1]. Inverse Problems, 2006, vol. 22, no. 4, pp. 1457–1484. DOI: https://doi.org/10.1088/0266-5611/22/4/018
  5. Gorbunov O. B., Shieh C.-T., Yurko V. A. Dirac system with a singularity in an interior point. Applicable Analysis, 2016, vol. 95, iss. 11, pp. 2397–2414. DOI: https://doi.org/10.1080/00036811.2015.1091069
  6. Beals R., Coifman R. R. Scattering and inverse scattering for first order systems. Comm. Pure Appl. Math., 1984, vol. 37, iss. 1, pp. 39–90. DOI: https://doi.org/10.1002/cpa.3160370105
  7. Zhou X. Direct and inverse scattering transforms with arbitrary spectral singularities. Comm. Pure Appl. Math. 1989, vol. 42, iss. 7, pp. 895–938. DOI: https://doi.org/10.1002/cpa.3160420702
  8. Yurko V. A. Inverse spectral problems for differential systems on a finite interval. Results Math., 2005, vol. 48, iss. 3–4, pp. 371–386. DOI: https://doi.org/10.1007/BF03323374
  9. Ignatyev M. Spectral analysis for differential systems with a singularity. Results Math., 2017, vol. 71, iss. 3–4, pp. 1531–1555. DOI: https://doi.org/10.1007/s00025-016-0605-0
  10. Yurko V. A. On higher-order differential operators with a singular point. Inverse Problems, 1993, vol. 9, no. 4, pp. 495–502. DOI: https://doi.org/10.1088/0266-5611/9/4/004
  11. Fedoseev A. E. Inverse problems for differential equations on the half-line having a singularity in an interior point. Tamkang Journal of Mathematics, 2011, vol. 42, no. 3, pp. 343–354. DOI: https://doi.org/10.5556/j.tkjm.42.2011.879
  12. Beals R., Deift P., Tomei C. Direct and inverse scattering on the line. Providence, Rhod Island, American Mathematical Society, 1988. 209 p.
  13. Sibuya Yu. Stokes phenomena. Bull. Amer. Math. Soc., 1977, vol. 83, no. 5, pp. 1075– 1077.
  14. Ignatiev M. Integral transforms connected with differential systems with a singularity. Tamkang Journal of Mathematics, 2019, vol. 50, no. 3, pp. 253–268. DOI: https://doi.org/10.5556/j.tkjm.50.2019.3353