A 2-edge partial inverse problem for the Sturm-Liouville operators with singular potentials on a star-shaped graph

Natalia Pavlovna Bondarenko

Abstract


Boundary value problems for Sturm-Liouville operators with potentials from the class $W_2^{-1}$ on a star-shaped graph are considered. We assume that the potentials are known on all the edges of the graph except two, and show that the potentials on the remaining edges can be constructed by fractional parts of two spectra. A uniqueness theorem is proved, and an algorithm for the constructive solution of the partial inverse problem is provided. The main ingredient of the proofs is the Riesz-basis property of specially constructed systems of functions.

Keywords


Partial Inverse Problem; Quantum Graph; Sturm-Liouville Operator; Singular Potential; Weyl Function; Riesz Basis

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References


P. Kuchment, Graph models for waves in thin structures, Waves in Random Media, 12(4)(2002), R1--R24.

Yu. V. Pokorny, O. M. Penkin and V. L. Pryadiev, et al., Differential Equations on Geometrical Graphs,Fizmatlit, Moscow (2004) (Russian).

V. A. Yurko, Inverse spectral problems for differential operators on spatial networks, Russian Mathematical Surveys, 71(2016), 539--584.

A. A. Shkalikov and A. M. Savchuk, Sturm-Liouville operators with singular potentials, Math.Notes, 66(1999), 741-753.

R. O. Hryniv and Ya. V. Mykytyuk, Inverse spectral problems for Sturm-Liouville operators with singular potentials, Inverse Problems, 19(2003), 665-684.

R. O. Hryniv and Ya. V. Mykytyuk, Inverse spectral problems for Sturm-Liouville operators with singular potentials, II.Reconstruction by two spectra, in: V. Kadets, W. Zelazko (Eds.), Functional Analysis and Its Applications, in: North-Holland Math. Stud., vol. 197, North-Holland Publishing, Amsterdam (2004), 97--114.

R. O. Hryniv and Ya. V. Mykytyuk, Half-inverse spectral problems for Sturm-Liouville operators with singular potentials, Inverse Problems, 20(2004), 1423--1444.

R. O. Hryniv and Ya. V. Mykytyuk, Transformation operators for Sturm-Liouville operators with singular potentials}, Math. Phys. Anal. Geom.,7(2004), 119--149.

G. Freiling, M. Ignatiev and V. Yurko, An inverse spectral problem for Sturm-Liouville operators with singular potentials on star-type graph, Proc. Symp. Pure Math., 77(2008), 397--408.

N. P. Bondarenko, Partial inverse problems for the Sturm-Liouville operator on a star-shaped graph with mixed boundary conditions, J. Inverse Ill-Posed Probl. (2017), DOI: 10.1515/jiip-2017-0001.

N. P. Bondarenko, A partial inverse problem for the Sturm-Liouville operator on a star-shaped graph, Anal. Math. Phys., (2017), DOI:10.1007/s13324-017-0172-x.

C.-F. Yang, Inverse spectral problems for the Sturm-Liouville operator on a $d$-star graph, J. Math. Anal. Appl., 365(2010), 742--749.

H. Hochstadt and B. Lieberman, An inverse Sturm-Liouville problem with mixed given data, SIAM J. Appl. Math., 34(1978), 676--680.

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

G. Freiling and V. Yurko, Inverse Sturm-Liouville Problems and Their Applications, Huntington,NY: Nova Science Publishers, 305 p. (2001).

S. A. Buterin, G. Freiling and V. A. Yurko, Lectures in the theory of entire functions, Schriftenriehe der Fakultat fur Matematik,Duisbug-Essen University, SM-UDE-779 (2014).

A. M. Sedletskii, Nonharmonic analysis, Journal of Mathematical Sciences, 116(5) (2003), 3551--3619.

G. G. Hardy and D. E. Littlewood and G. Polya, Inequalities, Cambridge Univ. Press, New York, 1934.

N. Bondarenko, and S. Buterin, On Recovering the Dirac Operator with an Integral Delay from the Spectrum, Results. Math., 71(3--4) (2017),1521--1529.




DOI: http://dx.doi.org/10.5556/j.tkjm.49.2018.2425

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