An inverse spectral problem for Sturm-Liouville operators with singular potentials on arbitrary compact graphs
Main Article Content
Abstract
Sturm-Liouville differential operators with singular potentials on arbitrary com- pact graphs are studied. The uniqueness of recovering operators from Weyl functions is proved and a constructive procedure for the solution of this class of inverse problems is provided.
Article Details
How to Cite
Vasiliev, S. V. (2019). An inverse spectral problem for Sturm-Liouville operators with singular potentials on arbitrary compact graphs. Tamkang Journal of Mathematics, 50(3), 293–305. https://doi.org/10.5556/j.tkjm.50.2019.3356
Issue
Section
Papers
References
[1] Marchenko V.A. Sturm-Liouville operators and their applications, ”Naukova Dumka”, Kiev, 1977; English transl., Birkhauser, 1986.
[2] Levitan B.M., Inverse Sturm-Liouville problems, Nauka, Moscow, 1984; English transl., VNU Sci.Press, Utrecht, 1987.
[3] Freiling G. and Yurko V. A. Inverse Sturm-Liouville Problems and their Applications : NOVA Science Publishers, New York, 2001. 305 p.
[4] Beals R., Deift P. and Tomei C. Direct and Inverse Scattering on the Line : Math. Surveys and Monographs v.28, RI, 1988. 252 p.
[5] Yurko V.A., Inverse Spectral Problems for Differential Operators and their Applications, Gordon and Breach, Amsterdam, 2000.
[6] Yurko V. A. Method of Spectral Mappings in the Inverse Problem Theory : Inverse and Illposed Problems Series, Utrecht, 2002. 303 p.
[7] Hryniv R. O. and Mykytyuk Ya. V. Inverse spectral problems for Sturm-Liouville operators with singular potentials : Inverse Problems 19, 2003. 665-684 p.
[8] Hryniv R. O. and Mykytyuk Ya. V. Transformation operators for Sturm-Liouville operators with singular potentials : Mathematical Physics, Analysis and Geometry 7, 2004. 119- 149 p.
[9] Shkalikov A.A. and Savchuk A.M., Sturm-Liouville operators with singular potentials, Math. Notes 66 (2003), 741-753.
[10] Pokornyi Yu.V. and Borovskikh A. V., Differential equations on networks (geometric graphs), J. Math. Sci. (N.Y.) 119, (2004), 691-718.
[11] Yu. Pokornyi and V. Pryadiev, The qualitative Sturm-Liouville theory on spatial networks, J. Math. Sci. (N.Y.) 119 (2004), 788-835.
[12] Belishev M. I. Boundary spectral inverse problem on a class of graphs (trees) by the BC method : Inverse Problems 20, 2004. 647-672 pp.
[13] Yurko V.A., Inverse spectral problems for Sturm-Liouville operators on graphs, Inverse Problems 21 (2005), 1075-1086.
[14] Bellmann R. and Coike K.L. Differential-difference Equations :Santa Monica, CA: RAND Corporation, 1963, 548p.
[15] Freiling G. and Yurko V.A. Inverse problems for differential operators on trees with general matching conditions : Applicable Analysis 86, no.6, 2007, 653-667pp.
[16] Yurko V.A. Inverse problems for Sturm-Liouville operators on graphs with a cycle. Oper- ators and Matrices, vol.2, no.4 (2008), 543-553.
[17] Yurko V. A. Inverse problem for SturmLiouville operators on hedgehog-type graphs. Math. Notes, 2011, vol. 89, no. 3, pp. 438-449.
[18] Yurko V.A. Inverse problems for Sturm-Liouville operators on bush-type graphs. : Inverse Problems 25, 2009, 125-127 pp.
[19] Yurko V.A. Uniqueness of recovering Sturm-Liouville operators on A-graphs from spectra. : Results in Mathematics 55, 2009, 199-207 pp.
[20] Yurko, V. A. On recovering SturmLiouville operators on graphs, Mat. Zametki, 2006, Volume 79, Issue 4, 619630
[21] Yurko V.A. Inverse spectral problems for differential operators on arbitrary compact graphs. Journal of Inverse and Ill-Posed Problems 18, no.3 (2010), 245-261.
[22] Yurko V.A. Inverse problems for differential of any order on trees. Matemat. Zametki 83,no.1 (2008), 139-152; English transl. in Math. Notes 83, no.1 (2008), 125-137.
[23] Stankevich I.V. An inverse problem of spectral analysis for Hill’s equation, Doklady Akad. Nauk SSSR, 192, no.1(1970), 34–37. (in Russian); English transl. in Soviet Math. Dokl., 11(1970), 582–586.
[24] Ignatiev M.Y. Inverse spectral problem for Sturm-Liouville operator on non-compact A- graph. Uniqueness result. : Tamkang Journal of Mathematics, 2013, 25 p.
[25] Ignatiev M.Y. Inverse scattering problem for Sturm-Liouville operator on one-vertex non- compact graph with a cycle. : Tamkan J. of Mathematics 42s, N3.011, 154-166pp.
[26] Conway J.B. Functions of One Complex Variable : vol.I, 2nd edn., Springer–Verlag, New York, 1995, 412p.
[27] Naimark M. A. Linear differential operators. Harrap, London ; Toronto, 1968.
[28] Evnin A. Yu., Polynomial as a sum of periodic functions, Vestn. Yuzhno-Ural. Gos. Un-ta. Ser. Matem. Mekh. Fiz., 2013.
[29] Vellucci P. A simple pointview for kadec-1/4 theorem in the complex case : Ricerche di Matematica, 2014, 87–92 p.
[30] Bondarenko N.P. A 2-edge partial inverse problem for the Sturm-Liouville operators with singular potentials on a star-shaped graph : Tamkang Journal of Mathematics, Vol. 49, No. 1., 2018, 49-66 pp.
[31] Bondarenko N.P. and Yang C.-F. A partial inverse problem for the Sturm-Liouville oper- ator on the lasso-graph : Inverse Problems and Imaging, Vol. 13, No. 1., 2019, 69 p.
[32] Bondarenko N.P. An inverse problem for Sturm-Liouville operators on trees with partial information given on the potentials, Mathematical Methods in the Applied Sciences (2019), Vol. 42, Issue 5, 1512-1528.
[33] Freiling G., Ignatiev M.Y. and Yurko V.A. An inverse spectral problem for Sturm-Liouville operators with singular potentials on star-type graph, Proc. Symp. Pure Math. 77 (2008), 397-408.
[2] Levitan B.M., Inverse Sturm-Liouville problems, Nauka, Moscow, 1984; English transl., VNU Sci.Press, Utrecht, 1987.
[3] Freiling G. and Yurko V. A. Inverse Sturm-Liouville Problems and their Applications : NOVA Science Publishers, New York, 2001. 305 p.
[4] Beals R., Deift P. and Tomei C. Direct and Inverse Scattering on the Line : Math. Surveys and Monographs v.28, RI, 1988. 252 p.
[5] Yurko V.A., Inverse Spectral Problems for Differential Operators and their Applications, Gordon and Breach, Amsterdam, 2000.
[6] Yurko V. A. Method of Spectral Mappings in the Inverse Problem Theory : Inverse and Illposed Problems Series, Utrecht, 2002. 303 p.
[7] Hryniv R. O. and Mykytyuk Ya. V. Inverse spectral problems for Sturm-Liouville operators with singular potentials : Inverse Problems 19, 2003. 665-684 p.
[8] Hryniv R. O. and Mykytyuk Ya. V. Transformation operators for Sturm-Liouville operators with singular potentials : Mathematical Physics, Analysis and Geometry 7, 2004. 119- 149 p.
[9] Shkalikov A.A. and Savchuk A.M., Sturm-Liouville operators with singular potentials, Math. Notes 66 (2003), 741-753.
[10] Pokornyi Yu.V. and Borovskikh A. V., Differential equations on networks (geometric graphs), J. Math. Sci. (N.Y.) 119, (2004), 691-718.
[11] Yu. Pokornyi and V. Pryadiev, The qualitative Sturm-Liouville theory on spatial networks, J. Math. Sci. (N.Y.) 119 (2004), 788-835.
[12] Belishev M. I. Boundary spectral inverse problem on a class of graphs (trees) by the BC method : Inverse Problems 20, 2004. 647-672 pp.
[13] Yurko V.A., Inverse spectral problems for Sturm-Liouville operators on graphs, Inverse Problems 21 (2005), 1075-1086.
[14] Bellmann R. and Coike K.L. Differential-difference Equations :Santa Monica, CA: RAND Corporation, 1963, 548p.
[15] Freiling G. and Yurko V.A. Inverse problems for differential operators on trees with general matching conditions : Applicable Analysis 86, no.6, 2007, 653-667pp.
[16] Yurko V.A. Inverse problems for Sturm-Liouville operators on graphs with a cycle. Oper- ators and Matrices, vol.2, no.4 (2008), 543-553.
[17] Yurko V. A. Inverse problem for SturmLiouville operators on hedgehog-type graphs. Math. Notes, 2011, vol. 89, no. 3, pp. 438-449.
[18] Yurko V.A. Inverse problems for Sturm-Liouville operators on bush-type graphs. : Inverse Problems 25, 2009, 125-127 pp.
[19] Yurko V.A. Uniqueness of recovering Sturm-Liouville operators on A-graphs from spectra. : Results in Mathematics 55, 2009, 199-207 pp.
[20] Yurko, V. A. On recovering SturmLiouville operators on graphs, Mat. Zametki, 2006, Volume 79, Issue 4, 619630
[21] Yurko V.A. Inverse spectral problems for differential operators on arbitrary compact graphs. Journal of Inverse and Ill-Posed Problems 18, no.3 (2010), 245-261.
[22] Yurko V.A. Inverse problems for differential of any order on trees. Matemat. Zametki 83,no.1 (2008), 139-152; English transl. in Math. Notes 83, no.1 (2008), 125-137.
[23] Stankevich I.V. An inverse problem of spectral analysis for Hill’s equation, Doklady Akad. Nauk SSSR, 192, no.1(1970), 34–37. (in Russian); English transl. in Soviet Math. Dokl., 11(1970), 582–586.
[24] Ignatiev M.Y. Inverse spectral problem for Sturm-Liouville operator on non-compact A- graph. Uniqueness result. : Tamkang Journal of Mathematics, 2013, 25 p.
[25] Ignatiev M.Y. Inverse scattering problem for Sturm-Liouville operator on one-vertex non- compact graph with a cycle. : Tamkan J. of Mathematics 42s, N3.011, 154-166pp.
[26] Conway J.B. Functions of One Complex Variable : vol.I, 2nd edn., Springer–Verlag, New York, 1995, 412p.
[27] Naimark M. A. Linear differential operators. Harrap, London ; Toronto, 1968.
[28] Evnin A. Yu., Polynomial as a sum of periodic functions, Vestn. Yuzhno-Ural. Gos. Un-ta. Ser. Matem. Mekh. Fiz., 2013.
[29] Vellucci P. A simple pointview for kadec-1/4 theorem in the complex case : Ricerche di Matematica, 2014, 87–92 p.
[30] Bondarenko N.P. A 2-edge partial inverse problem for the Sturm-Liouville operators with singular potentials on a star-shaped graph : Tamkang Journal of Mathematics, Vol. 49, No. 1., 2018, 49-66 pp.
[31] Bondarenko N.P. and Yang C.-F. A partial inverse problem for the Sturm-Liouville oper- ator on the lasso-graph : Inverse Problems and Imaging, Vol. 13, No. 1., 2019, 69 p.
[32] Bondarenko N.P. An inverse problem for Sturm-Liouville operators on trees with partial information given on the potentials, Mathematical Methods in the Applied Sciences (2019), Vol. 42, Issue 5, 1512-1528.
[33] Freiling G., Ignatiev M.Y. and Yurko V.A. An inverse spectral problem for Sturm-Liouville operators with singular potentials on star-type graph, Proc. Symp. Pure Math. 77 (2008), 397-408.