Necessary and sufficient conditions for the solvability of the inverse problem for non-self-adjoint pencils of Sturm-Liouville operators on the half-line

Authors

  • Vjacheslav Yurko Department of Mathematics, Saratov University

DOI:

https://doi.org/10.5556/j.tkjm.42.2011.742

Keywords:

Sturm-Liouville operators, boundary conditions with the spectral parameter, inverse spectral problems, method of spectral mappings

Abstract

Non-self-adjoint Sturm-Liouville differential operators on the half-line with a boundary condition depending polynomially on the spectral parameter are studied. We investigate the inverse problem of recovering the operator from the Weyl function. For this inverse problem we provide necessary and suffcient conditions for its solvability along with a procedure for constructing its solution by the method of spectral mappings.

Author Biography

Vjacheslav Yurko, Department of Mathematics, Saratov University

Department of Mathematics

Saratov University

References

L. Collatz, Eigenwertaufgaben mit technischen Anwendungen, Akad. Verlagsgesellschaft Geest & Portig, Leipzig, 1963.

R. Mennicken and M. M"oller, Non-self-adjoint boundary value problems. North-Holland Mathematic Studies, vol. 192, Amsterdam, North-Holland, 2003.

A. A. Shkalikov, Boundary problems for opdinary problems for differential equations with parameter in the boundary conditions, J. Sov. Math., 33 (1986), 1311-1342; translation from Tr. Semin. im. I.G. Petrovskogo, 9(1983), 190--229.

Ch. Tretter, Boundary eigenvalue problems with differential equations $Neta=lambda Peta$ with $lambda$-- polynomial boundary conditions, J. Differ. Equ., 170 (2001), 408--471.

V. A. Marchenko, Sturm-Liouville operators and their applications, Naukova Dumka, Kiev, 1977; English transl., Birkh"auser, 1986.

B. M. Levitan, Inverse Sturm-Liouville Problems, Nauka, Moscow, 1984 (Russian); English transl., VNU Sci.Press, Utrecht, 1987.

J. P"oschel and E. Trubowitz, Inverse Spectral Theory, New York, Academic Press, 1987.

G. Freiling and V. A. Yurko, Inverse Sturm-Liouville Problems and their Applications. NOVA Science Publishers, New York, 2001.

V. A. Yurko, Method of Spectral Mappings in the Inverse Problem Theory, Inverse and Ill-posed Problems Series. VSP, Utrecht, 2002.

A. G. Ramm, Inverse problems. Mathematical and Analytical Technique with Applications to Engineering. Springer, New York, 2005.

P. J. Browne and B. D. Sleeman, A uniqueness theorem for inverse eigenparameter dependent Sturm-Liouville problems, Inverse Problems, 13 (1997), 1453--1462.

M. V. Chugunova, Inverse spectral problem for the Sturm-Liouville operator with eigenvalue parameter dependent boundary conditions, Oper. Theory: Advan. Appl., 123, Birkhauser, Basel, 2001, 187--94.

P. A. Binding, P. J. Browne and B. A. Watson, Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter, II. J. Comp. Appl. Math., 148 (2002), 147--168.

P. A. Binding, P. J. Browne and B. A. Watson, Equivalence of inverse Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter, J. Math. Anal. Appl., 291 (2004), 246--261.

N. J. Guliyev, Inverse eigenvalue problems for Sturm-Liouville equations with spectral parameter linearly contained in one of the boundary condition, Inverse Problems, 21 (2005), 1315-1330.

G. Freiling and V. A. Yurko, Inverse problems for Sturm-Liouville equations with boundary conditions polynomially dependent on the spectral parameter, Inverse Problems, 26(2010), 055003, 17pp.

Yurko V.A. Recovering non-self-adjoint Sturm-Liouville pencils on the half-line. Schriftenreiche des Fachbereichs Mathematik, SM-DU-726, Universitaet Duisburg-Essen, 2011, 10pp.

M. Ignatiev and V. A. Yurko, Numerical methods for solving inverse Sturm-Liouville problems, Results in Math., 52 (2008), 63--74.

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Published

2011-08-24

How to Cite

Yurko, V. (2011). Necessary and sufficient conditions for the solvability of the inverse problem for non-self-adjoint pencils of Sturm-Liouville operators on the half-line. Tamkang Journal of Mathematics, 42(3), 247–258. https://doi.org/10.5556/j.tkjm.42.2011.742

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