Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

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


For citation:

Efremova L. S. Numerical Solution of Inverse Spectral Problems for Sturm–Liouville Operators with Discontinuous Potentials. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2014, vol. 14, iss. 3, pp. 273-279. DOI: 10.18500/1816-9791-2014-14-3-273-279, EDN: SMSJUR

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
10.09.2014
Full text:
(downloads: 125)
Language: 
Russian
Heading: 
UDC: 
517.984
EDN: 
SMSJUR

Numerical Solution of Inverse Spectral Problems for Sturm–Liouville Operators with Discontinuous Potentials

Autors: 
Efremova Liubov Sergeevna, Saratov State University
Abstract: 

We consider Sturm–Liouville differential operator with potential having a finite number of simple discontinuities. This paper is devoted to the numerical solution of such inverse spectral problems. The main result of this work is a procedure that is able to recover both the points of discontinuities as well as the heights of the jumps. Following, using these results, we may apply a suitable numerical method (for example, the generalized Rundell–Sacks algorithm with a special form of the reference potential) to reconstruct the potential more precisely.

References: 
  1. Levitan B. M. Inverse Sturm– Liouville Problems.Utrecht, VNU Sci. Press, 1987, 240 p.
  2. Marchenko V. A. Sturm– Liouville operators and applications. Basel, Birkh¨auser, 1986. 367 p.
  3. Ignatiev M. Yu., Yurko V. A. Numerical methods for solving inverse Sturm– Liouville problems. Results in Math., 2008, vol. 52, pp. 63–74. DOI: 10.1007/s00025-007-0276-y.
  4. Rafler M., B¨ockmann C. Reconstruction method for inverse Sturm– Liouville problems with discontinuous potentials. Inverse Problems, 2007, vol. 23, no. 3, pp. 933–946. DOI: 10.1088/0266-5611/23/3/006.
  5. Rundell W., Sacks P. E. Reconstruction techniques for classical inverse Sturm– Liouville problems. Mathematics of Computation, 1992, vol. 58, no. 197, pp. 161–183. DOI: 10.1090/S0025-5718-1992-1106979-0.
  6. Freiling G., Yurko V. A. Inverse Sturm– Liouville Problems and Their Applications. Huntington, New York, NOVA Science Publ., 2001, 305 p.
  7. Oppenheim A. V., Schafer R. W. Discrete-time Signal Processing. Prentice-Hall, 1975, 585 p.
  8. Vinokurov V. A., Sadovnichii V. A. Asymptotics of any order for the eigenvalues and eigenfunctions of the Sturm– Liouville boundary-value problem on a segment with a summable potential. Izvestiya :Mathematics, 2000, vol. 64, iss. 4, pp. 695–754. DOI: http://dx.doi.org/10.4213/im295.
Received: 
13.03.2014
Accepted: 
18.07.2014
Published: 
10.09.2014