An inverse problem for the non-self-adjoint matrix Sturm-Liouville operator

Natalia Pavlovna Bondarenko

Abstract


The inverse problem of spectral analysis for the non-self-adjoint matrix Sturm-Liouville operator on a finite interval is investigated. We study properties of the spectral characteristics for the considered operator, and provide necessary and sufficient conditions for the solvability of the inverse problem. Our approach is based on the constructive solution of the inverse problem by the method of spectral mappings. The characterization of the spectral data in the self-adjoint case is given as a corollary of the main result.

Keywords


Matrix Sturm-Liouville Equation; inverse Spectral Problems; Necessary and Sufficient Conditions; Method of Spectral Mappings

Full Text:

PDF

References


V. A. Marchenko, Sturm-Liouville Operators and their Applications, Naukova Dumka, Kiev (1977) (Russian); English transl., Birkhauser, 1986.

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

J. Poschel and E. Trubowitz, Inverse Spectral Theory, New York, Academic Press, 1987.

G. Freiling and V. Yurko, Inverse Sturm-Liouville problems and their applications, Huntington, NY: Nova Science Publishers, 305 p., 2001.

Z. S. Agranovich and V. A. Marchenko, The inverse problem of scattering theory [in Russian], KSU, Kharkov, 1960; Gordon and Breach, New York, 1963 (Eng. Transl.).

R. Carlson, An inverse problem for the matrix Schrodinger equation, J. Math. Anal. Appl., 267(2002), 564--575.

M. M. Malamud, Uniqueness of the matrix Sturm-Liouville equation given a part of the monodromy matrix, and Borg type results, Sturm-Liouville Theory, Birkhauser, Basel, (2005), 237--270.

V. A. Yurko, Inverse problems for matrix Sturm-Liouville operators, Russian J. Math. Phys., 13,no.1 (2006), 111--118.

V. A. Yurko, Inverse problems for the matrix Sturm-Liouville equation on a finite interval, Inverse Problems, 22(2006), 1139--1149.

D. Chelkak and E. Korotyaev, Weyl-Titchmarsh functions of vector-valued Sturm-Liouville operators on the unit interval, J. Func. Anal. 257(2009), 1546--1588.

N. Bondarenko, Spectral analysis for the matrix Sturm-Liouville operator on a finite interval, Tamkang J. Math., 42: 3 (2011), 305--327.

Ya. V. Mykytyuk and N. S. Trush, Inverse spectral problems for Sturm-Liouville operators with matrix-valued potentials, Inverse Problems, 26(2010), 015009.

N. Bondarenko, An inverse spectral problem for the matrix Sturm-Liouville operator on the half-line, Boundary Value Problems. 2015:15.

N. Bondarenko, Inverse scattering on the line for the matrix Sturm-Liouville equation, J. Differential Equations, 262(2017),Issue 3, 2073--2105.

V. A. Yurko, An inverse spectral problem for singular non-self-adjoint differential systems, Sbornik: Mathematics (2004) 195:12, 1823--1854.

V. A. Yurko, Reconstruction of non-self-adjoint differential systems on the half-Line from the Weyl matrix, Mathematical Notes (2004), 76, no.~2, 296--302.

V. A. Yurko, An inverse problem for differential systems on a finite interval in the case of multiple roots of the characteristic polynomial, Diff. Equ., 41(2005), no. 6, 818--823.

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

S. A. Buterin, On inverse spectral problem for non-selfadjoint Sturm-Liouville operator on a finite interval, Journal of Mathematical Analysis and Applications, 335(2007), Issue 1, 739--749.

S. A. Buterin, C.-T. Shieh, and V. A. Yurko, Inverse spectral problems for non-selfadjoint second-order differential operators with Dirichlet boundary conditions,

Boundary Value Problems (2013), 2013:180, 1--24.

M. A. Naimark, Linear Differential Operators, 2nd ed., Nauka, Moscow (1969); English transl. of 1st ed., Parts I,II, Ungar, New York (1967, 1968).




DOI: http://dx.doi.org/10.5556/j.tkjm.50.2019.2735

Sponsored by Tamkang University | ISSN 0049-2930 (Print), ISSN 2073-9826 (Online) | Powered by MathJax