Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

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

For citation:

Yurko V. A. Recovering singular differential pencils with a turning point. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2005, vol. 5, iss. 1, pp. 71-81. DOI: 10.18500/1816-9791-2005-5-1-71-81, EDN: PPDABA

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
30.09.2005
Full text:
download
(downloads: 178)
Language: 
English
Heading: 
Mathematics
UDC: 
517.95
DOI: 
10.18500/1816-9791-2005-5-1-71-81
EDN: 
PPDABA

Recovering singular differential pencils with a turning point

Autors: 
Yurko Vjacheslav Anatol'evich, Saratov State University
Abstract: 

Second-order pencils of differential equations on the half-line with turning points are considered. We establish properties of the spectrum and study the inverse spectral problem of recovering coefficients of the pencil from the spectral data.

Key words: 
Singular differential pencil
spectrum
inverse problem
References: 
  1. Tamarkin J. D., On some problems of the theory of ordinary linear differential equations, Petrograd, 1917
  2. Keldysh M. V., “On eigenvalues and eigenfunctions of some classes of nonselfadjoint equations”, Dokl. Akad. Nauk SSSR, 77 (1951), 11–14 Z
  3. McHugh J., “An historical survey of ordinary linear differential equations with а large parameter and turning points”, Arch. Hist. Exact. Sci., 7 (1970), 277–324
  4. Kostyuchenko A. G., Shkalikov A. A., “Selfadjoint quadratic operator pencils and elliptic problems”, Funkt. Anal. Prilozhen., 17:2 (1983), 38–61 (Russian) 
  5. Freiling G., “On the completeness and minimality of the derived chains of eigen and associated functions of boundary eigenvalue problems nonlinearly dependent on the parameter”, Results in Math., 14 (1988), 64–83
  6. Wasow W., Linear turning point theory, Berlin, 1985
  7. Eberhard W., Freiling G., “An expansion theorem for eigenvalue problems with several turning points”, Analysis, 13 (1993), 301–308
  8. Beals R., “Indefinite Sturm–Liouville problems and half-range completeness”, J. Diff. Equations, 56:3 (1985), 391–407
  9. Langer Н., Curgus В., “A Кrein space approach to symmetric ordinary differential operators with an indefinite weight function”, J. Diff. Equations, 79:1 (1989), 31–61
  10. Marchenko V. A., Sturm–Liouville operators and their applications, Кiev, 1977 (Russian)
  11. Levitan В. М., Inverse Sturm–Liouville problems, М., 1984 (Russian); Utrecht, 1987
  12. Freiling G.. Yurko V. A., Inverse Sturm–Liouville proЬlems and their applications, N.Y., 2001
  13. Gasymov М. G., Gusejnov G. S., “Determination of diffusion operators according to spectral data”, Dokl. Akad. Nauk Az. SSR, 37:2 (1981), 19–23
  14. Yamamoto М., “Inverse eigenvalue problem for а vibration of а string with viscous drag”, J. Math. Anal. Appl., 152:1 (1990), 20–34
  15. Khruslov E. Y., Shepelsky D. G., “Inverse scattering method in electromagnetic sounding theory”, Inverse ProЬlems, 10:1 (1994), 1–37
  16. Yurko V. A., “An inverse problem for systems of differential equations with nonlinear dependence on the spectral parameter”, Diff. Uravneniya, 33:3 (1997), 390–395 (Russian)
  17. Aktosun Т., Кlaus М., Мее С. van der, “Inverse scattering in one-dimensional nonconservative media”, Integral Equat. Oper. Theory, 30:3 (1998), 279–316
  18. Pivovarchik V., “Reconstruction of the potential of the Sturm–Liouville equation from three spectra of boundary value problems”, Funct. Anal. i Prilozh., 33:3 (1999), 87–90 (Russian)
  19. Yurko V. A., “An inverse problem for pencils of differential operators”, Mat. Sbornik, 191:10 (2000), 137–160 (Russian) 
  20. Belishev M. I., “An inverse spectral indefinite problem for the equation y + λr(x)y = O on an interval”, Funct. Anal. i Prilozh., 21:2 (1987), 68–69 (Russian) 
  21. Darwish A. A., “On the inverse scattering problem for а generalized Sturm–Liouville differential operator”, Kyungpook Math. J., 29:1 (1989), 87–103
  22. El-Reheem, Zaki F. A., “The inverse scattering problem for some singular Sturm– Liouville operator”, Pure Math. Appl., 8:2–4 (1997), 233–246
  23. Freiling G., Yurko V. A., “Inverse problems for differential equations with turning points”, Inverse Problems, 13 (1997), 1247–1263
  24. Freiling G., Yurko V. A., “Inverse spectral problems for differential equations on the half-line with turning points”, J. Diff. Equations, 154 (1999), 419–453
  25. Bennewitz С., “A Paley–Wiener theorem with applications to inverse spectral theory”, Advances in diff. equations and math. physics, Birmingham, AL, 2002, 21–31
  26. Yurko V. A., Method of spectral mappings in the inverse problem theory, Inverse and ill-posed problems series, Utrecht, 2002
  27. Coddington Е., Levinson N., Theory of ordinary differential equations, N.Y., 1955
  28. Rykhlov V. S., “Asymptotical formulas for solutions of linear differential systems of the first order”, Results Math., 36:3–4 (1999), 342–353
  29. Mennicken R., Moeller М., Non-self-adjoint boundary eigenvalue problems, Amsterdam, 2003
  30. Levitan В. М., Sargsjan I. S., Introduction to spectral theory, М., 1970 (Russian); AMS Transl. of Math. Monographs, 39, Providence, RI, 1975 (English)
  31. Leibenzon Z. L., “The inverse problem of spectral analysis for higher-order ordinary differential operators”, Trudy Mosk. Mat. Obshch., 15, 1966, 70–144 (Russian)
Received: 
02.03.2005
Accepted: 
29.08.2005
Published: 
30.09.2005
  • 1010 reads