Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Inverse problem of acoustic scattering for centrally symmetric finite objects in two-dimensional space


Scattering theory for the wave equation in two-dimensional space, perturbed by a finite function of a radial variable, integrable everywhere except, perhaps, the origin of coordinates, is considered from the point of view of the LaxPhillips scheme. The compression operator, related to the corresponding scattering problem, is considered. It is shown that this compression has one-dimensional defect subspaces, and its characteristic operator-function is a meromorphic function, whose zeros and poles coincide, respectively, with the corresponding values of a dissipative operator and its adjoint. The solution of the inverse scattering problem is obtained by reducing it to the inverse problem with two spectra for the singular self-adjoint Sturm-Liouville operator.

This is a preview of subscription content, log in to check access.

Literature cited

  1. 1.

    P. D. Lax and R. S. Phillips, Scattering Theory, 2nd edn., Academic Press, Boston (1989).

  2. 2.

    P. D. Lax and R. S. Phillips, “Scattering theory for acoustic equation in an even number of space dimensions,” Indiana Univ. Math. J.,22, 101–134 (1972).

  3. 3.

    V. M. Adamyan, “Scattering theory for wave equations in even-dimensional spaces,” Funkts. Anal. Prilozhen.,10, No. 4, 1–8 (1976).

  4. 4.

    M. Reed and B. Simon, Methods of Modern Mathematical Physics, Academic Press, New York (1980).

  5. 5.

    V. I. Smirnov, Course of Higher Mathematics [in Russian], Vol. 4, Gostekhteorizdat, Moscow (1958).

  6. 6.

    M. G. Gasymov and B. M. Levitan, “Differential Sturm-Liouville operators with a discrete spectrum,” Mat. Sb.,63, No. 3, 445–458 (1964).

Download references

Author information

Additional information

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 12, pp. 1649–1657, December, 1990.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Mil'man, A.L. Inverse problem of acoustic scattering for centrally symmetric finite objects in two-dimensional space. Ukr Math J 42, 1484–1491 (1990).

Download citation


  • Inverse Problem
  • Wave Equation
  • Meromorphic Function
  • Radial Variable
  • Scattering Problem