Recovering singular differential operators on noncompact star-type graphs from Weyl functions

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Published: Jun 1, 2011
DOI: https://doi.org/10.5556/j.tkjm.42.2011.928
Keywords:
Singular differential equations noncompact graphs inverse spectral problems method of spectral mappings.

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V. Yurko
Department of Mathematics, Saratov State University

Abstract

Bessel-type differential operators on noncompact star-type graphs are studied. We establish properties of the spectral characteristics and then we investigate the inverse problem of recovering the operator from the so-called Weyl vector. For this inverse problem we prove a uniqueness theorem and propose a procedure for constructing the solution using the method of spectral mappings.

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How to Cite
Yurko, V. (2011). Recovering singular differential operators on noncompact star-type graphs from Weyl functions. Tamkang Journal of Mathematics, 42(2), 223–236. https://doi.org/10.5556/j.tkjm.42.2011.928
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Vol. 42 No. 2 (2011)
Section
Papers
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This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

References

M. D. Faddeev and B. S. Pavlov, Model of free electrons and the scattering problem, Teor. Math. Fiz., 55, (1983), 257–269 (Russian); English transl. in Theor.Math. Phys., 55 (1983), 485–492.

J. E. Langese, G. Leugering and J. P. G. Schmidt, Modelling, Analysis and Control of Dynamic Elastic Multi-Link Structures, Birkhäuser, Boston, 1994.

T. Kottos and U. Smilansky, Quantumchaos on graphs, Phys. Rev. Lett., 79 (1997), 4794–4797.

A. Sobolev and M. Solomyak, Schrödinger operator on homogeneous metric trees: spectrum in gaps, Rev. Math. Phys., 14, (2002), 421–467.

Yu. V. Pokornyi and A. V. Borovskikh, Differential equations on networks (geometric graphs), J. Math. Sci.

(N.Y.), 119, (2004), 691–718.

Yu. Pokornyi and V. Pryadiev, The qualitative Sturm-Liouville theory on spatial networks, J.Math. Sci. (N.Y.), 119 (2004), 788–835.

M. I. Belishev, Boundary spectral inverse problem on a class of graphs (trees) by the BC method, Inverse Problems 20, (2004), 647–672.

V. A. Yurko, Inverse spectral problems for Sturm-Liouville operators on graphs, Inverse Problems 21 (2005), 1075–1086.

B. M. Brown and R. Weikard, A Borg-Levinson theorem for trees, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461, (2005), 3231–3243.

P. Kurasov and M. Nowaczyk, Inverse spectral problem for quantum graphs, J. Phys. A, 38 (2005), 4901–4915.

V. Pivovarchik, Inverse problem for the Sturm-Liouville equation on a simple graph, SIAM J. Math. Anal., 32,

(2000), 801–819.

V. A. Yurko, Inverse problems for Sturm-Liouville operators on bush-type graphs, Inverse Problems 25, (2009), 105008, 14pp.

V. A. Yurko, Uniqueness of recovering Sturm-Liouville operators on A-graphs from spectra, Results in Mathematics, 55, (2009), 199–207.

V. A. Yurko, An inverse problem for Sturm-Liouville operators on arbitrary compact spatial networks, Dok-lady Akad. Nauk, 432, (2010), 318–321; English transl: Doklady Mathematics 81, (2010), 410–413.

N. I. Gerasimenko, Inverse scattering problem on a noncompact graph, Teoret. Mat. Fiz., 74 (1988), 187-200 (Russian); English transl. in Theor.Math. Phys., 75 (1988), 460–470.

G. Freiling and V. A. Yurko, Inverse spectral problems for Sturm-Liouville operators on noncompact trees,Results in Math., 50 (2007), 195–212.

V. Pivovarchik and Y. Latushkin, Scattering in a forked-shaped waveguide, Integral Equat. Oper. Theory 61 (2008), 365–399.

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

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

R. Beals, P. Deift and C. Tomei, Direct and Inverse Scattering on the Line, Math. Surveys and Monographs, v.28. Amer.Math. Soc. Providence: RI, 1988.

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

J. R. McLaughlin, Analytical methods for recovering coefficients in differential equations from spectral data, SIAM Rev., 28 (1986), 53–72.

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

New York, 2001.

K. Chadan, D. Colton, L. Paivarinta and W. Rundell, An introduction to inverse scattering and inverse spectral problems, SIAM Monographs on Mathematical Modeling and Computation, SIAM, Philadelphia, PA, 1997.

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

V. A. Yurko, Inverse Spectral Problems for Differential Operators and their Applications, Gordon and Breach, Amsterdam, 2000, 253pp.

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

B. M. Levitan and I. S. Sargsyan, Introduction to Spectral Theory, AMS Transl. of Math. Monogr. 39, Providence, 1975.

G. Freiling and V. A. Yurko, Inverse problems for differential operators with singular boundary conditions, Mathematishe Nachrichten 278, (2005), 1561-1578.

V. A. Yurko, Inverse problemfor differential equations with a singularity, Differ.Uravneniya, 28, (1992), 1355–1362 (Russian); English transl. in Diff. Equations, 28 (1992), 1100-1107.