# American Institute of Mathematical Sciences

February  2019, 13(1): 69-79. doi: 10.3934/ipi.2019004

## A partial inverse problem for the Sturm-Liouville operator on the lasso-graph

 1 Department of Applied Mathematics, Nanjing University of Sciences and Technology, Nanjing 210094, Jiangsu, China 2 Department of Applied Mathematics and Physics, Samara National Research University, Moskovskoye Shosse 34, Samara 443086, Russia 3 Department of Mechanics and Mathematics, Saratov State University, Astrakhanskaya 83, Saratov 410012, Russia

Received  November 2017 Revised  August 2018 Published  December 2018

The Sturm-Liouville operator with singular potentials on the lasso graph is considered. We suppose that the potential is known a priori on the boundary edge, and recover the potential on the loop from a part of the spectrum and some additional data. We prove the uniqueness theorem and provide a constructive algorithm for the solution of this partial inverse problem.

Citation: Chuan-Fu Yang, Natalia Pavlovna Bondarenko. A partial inverse problem for the Sturm-Liouville operator on the lasso-graph. Inverse Problems & Imaging, 2019, 13 (1) : 69-79. doi: 10.3934/ipi.2019004
##### References:
 [1] G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, Amer. Math. Soc., Providence, RI, 2013.  Google Scholar [2] N. Bondarenko and S. Buterin, On a local solvability and stability of the inverse transmission eigenvalue problem, Inverse Problems, 33 (2017), 115010, 19pp. doi: 10.1088/1361-6420/aa8cb5.  Google Scholar [3] N. P. Bondarenko, A partial inverse problem for the Sturm-Liouville operator on a star-shaped graph, Anal. Math. Phys., 8 (2018), 155-168.  doi: 10.1007/s13324-017-0172-x.  Google Scholar [4] 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., 26 (2018), 1-12.  doi: 10.1515/jiip-2017-0001.  Google Scholar [5] N. P. Bondarenko, A 2-edge partial inverse problem for the Sturm-Liouville operators with singular potentials on a star-shaped graph, Tamkang J. Math., 49 (2018), 49-66.  doi: 10.5556/j.tkjm.49.2018.2425.  Google Scholar [6] 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.  doi: 10.1090/pspum/077/2459883.  Google Scholar [7] G. Freiling and V. Yurko, Inverse Sturm-Liouville problems and their applications, Nova Science Publishers, Inc., Huntington, NY, 2001.  Google Scholar [8] X. He and H. Volkmer, Riesz bases of solutions of Sturm-Liouville equations, J. Fourier Anal. Appl., 7 (2001), 297-307.  doi: 10.1007/BF02511815.  Google Scholar [9] H. Hochstadt and B. Lieberman, An inverse Sturm-Liouville problem with mixed given data, SIAM J. Appl. Math., 34 (1978), 676-680.  doi: 10.1137/0134054.  Google Scholar [10] R. O. Hryniv and Ya. V. Mykytyuk, Transformation operators for Sturm-Liouville operators with singular potentials, Math. Phys. Anal. Geom., 7 (2004), 119-149.  doi: 10.1023/B:MPAG.0000024658.58535.74.  Google Scholar [11] R. O. Hryniv and Ya. V. Mykytyuk, Inverse spectral problems for Sturm-Liouville operators with singular potentials, Inverse Problems, 19 (2003), 665-684.  doi: 10.1088/0266-5611/19/3/312.  Google Scholar [12] 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, NorthHolland Publishing, Amsterdam, (2004), 97-114. doi: 10.1016/S0304-0208(04)80159-2.  Google Scholar [13] R. O. Hryniv and Ya. V. Mykytyuk, Half-inverse spectral problems for Sturm-Liouville operators with singular potentials, Inverse Problems, 20 (2004), 1423-1444.  doi: 10.1088/0266-5611/20/5/006.  Google Scholar [14] P. Kuchment, Quantum graphs. Some basic structures, Waves Random Media, 14 (2004), S107-S128.  doi: 10.1088/0959-7174/14/1/014.  Google Scholar [15] P. Kuchment, Graph models for waves in thin structures, Waves in Random Media, 12 (2002), R1-R24.  doi: 10.1088/0959-7174/12/4/201.  Google Scholar [16] P. Kurasov, Inverse scattering for lasso graph, J. Math. Phys., 54 (2013), 04210314, 14pp. doi: 10.1063/1.4799034.  Google Scholar [17] B. M. Levitan, Inverse Sturm-Liouville Problems, Nauka, Moscow, 1984 (Russian); English transl., VNU Sci. Press, Utrecht, 1987.  Google Scholar [18] V. A. Marchenko, Sturm-Liouville Operators and their Applications, Naukova Dumka, Kiev, 1977 (Russian); English transl., Birkhauser, 1986.  Google Scholar [19] V. Marchenko, K. Mochizuki and I. Trooshin, Inverse scattering on a graph, containing circle, Analytic Methods of Analysis and Differ, Equations: AMADE 2006, 237-243. Cambridge Sci. Publ., Cambridge, 2008.  Google Scholar [20] V. A. Marchenko and I. V. Ostrovskii, A characterization of the spectrum of the Hill operator, Mat. Sbornik, 97 (1975), 540-606 (Russian); English transl. in Math. USSR Sbornik, 26 (1975), 493-554.  Google Scholar [21] K. Mochizuki and I. Trooshin, On the scattering on a loop shaped graph, Evolution Equations of hyperbolic and Schroedinger Type, 227-245, Progr. Math., 301, Birkhauser/Springer. Basel A6, Basel, 2012. doi: 10.1007/978-3-0348-0454-7_12.  Google Scholar [22] V. N. Pivovarchik, Inverse problem for the Sturm-Liouville equation on a simple graph, SIAM J. Math. Anal., 32 (2000), 801-819.  doi: 10.1137/S0036141000368247.  Google Scholar [23] Yu. V. Pokorny, O. M. Penkin and V. L. Pryadiev, et al., Differential equations on geometrical graphs, Fizmatlit, Moscow, 2004 (Russian). Google Scholar [24] J. Pöschel and E. Trubowitz, Inverse Spectral Theory, New York, Academic Press, 1987.  Google Scholar [25] A. M. Savchuk, On the eigenvalues and eigenfunctions of the Sturm-Liouville operator with a singular potential, Mathematical Notes, 69 (2001), 245-252.  doi: 10.1023/A:1002880520696.  Google Scholar [26] I. V. Stankevich, An inverse problem of spectral analysis for Hill's equations, Doklady Akad. Nauk SSSR, 192 (1970), 34-37 (Russian).   Google Scholar [27] C.-F. Yang, Inverse spectral problems for the Sturm-Liouville operator on a $d$-star graph, J. Math. Anal. Appl., 365 (2010), 742-749.  doi: 10.1016/j.jmaa.2009.12.016.  Google Scholar [28] C.-F. Yang and X.-P. Yang, Uniqueness theorems from partial information of the potential on a graph, J. Inverse Ill-Posed Prob., 19 (2011), 631-641.  doi: 10.1515/JIIP.2011.059.  Google Scholar [29] V. A. Yurko, Inverse nodal problems for the Sturm-Liouville differential operators on a star-type graph, Siberian Math. J., 50 (2009), 373-378.  doi: 10.1007/s11202-009-0043-2.  Google Scholar [30] V. A. Yurko, Inverse problems for Sturm-Liouville operators on graphs with a cycle, Operators and Matrices, 2 (2008), 543-553.  doi: 10.7153/oam-02-34.  Google Scholar [31] V. A. Yurko, Inverse spectral problems for differential operators on spatial networks, Russian Mathematical Surveys, 71 (2016), 539-584.  doi: 10.4213/rm9709.  Google Scholar [32] V. A. Yurko, Inverse spectral problems for differential operators on arbitrary compact graphs, J. Inverse and Ill-Posed Probl., 18 (2010), 245-261.  doi: 10.1515/JIIP.2010.009.  Google Scholar

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##### References:
 [1] G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, Amer. Math. Soc., Providence, RI, 2013.  Google Scholar [2] N. Bondarenko and S. Buterin, On a local solvability and stability of the inverse transmission eigenvalue problem, Inverse Problems, 33 (2017), 115010, 19pp. doi: 10.1088/1361-6420/aa8cb5.  Google Scholar [3] N. P. Bondarenko, A partial inverse problem for the Sturm-Liouville operator on a star-shaped graph, Anal. Math. Phys., 8 (2018), 155-168.  doi: 10.1007/s13324-017-0172-x.  Google Scholar [4] 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., 26 (2018), 1-12.  doi: 10.1515/jiip-2017-0001.  Google Scholar [5] N. P. Bondarenko, A 2-edge partial inverse problem for the Sturm-Liouville operators with singular potentials on a star-shaped graph, Tamkang J. Math., 49 (2018), 49-66.  doi: 10.5556/j.tkjm.49.2018.2425.  Google Scholar [6] 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.  doi: 10.1090/pspum/077/2459883.  Google Scholar [7] G. Freiling and V. Yurko, Inverse Sturm-Liouville problems and their applications, Nova Science Publishers, Inc., Huntington, NY, 2001.  Google Scholar [8] X. He and H. Volkmer, Riesz bases of solutions of Sturm-Liouville equations, J. Fourier Anal. Appl., 7 (2001), 297-307.  doi: 10.1007/BF02511815.  Google Scholar [9] H. Hochstadt and B. Lieberman, An inverse Sturm-Liouville problem with mixed given data, SIAM J. Appl. Math., 34 (1978), 676-680.  doi: 10.1137/0134054.  Google Scholar [10] R. O. Hryniv and Ya. V. Mykytyuk, Transformation operators for Sturm-Liouville operators with singular potentials, Math. Phys. Anal. Geom., 7 (2004), 119-149.  doi: 10.1023/B:MPAG.0000024658.58535.74.  Google Scholar [11] R. O. Hryniv and Ya. V. Mykytyuk, Inverse spectral problems for Sturm-Liouville operators with singular potentials, Inverse Problems, 19 (2003), 665-684.  doi: 10.1088/0266-5611/19/3/312.  Google Scholar [12] 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, NorthHolland Publishing, Amsterdam, (2004), 97-114. doi: 10.1016/S0304-0208(04)80159-2.  Google Scholar [13] R. O. Hryniv and Ya. V. Mykytyuk, Half-inverse spectral problems for Sturm-Liouville operators with singular potentials, Inverse Problems, 20 (2004), 1423-1444.  doi: 10.1088/0266-5611/20/5/006.  Google Scholar [14] P. Kuchment, Quantum graphs. Some basic structures, Waves Random Media, 14 (2004), S107-S128.  doi: 10.1088/0959-7174/14/1/014.  Google Scholar [15] P. Kuchment, Graph models for waves in thin structures, Waves in Random Media, 12 (2002), R1-R24.  doi: 10.1088/0959-7174/12/4/201.  Google Scholar [16] P. Kurasov, Inverse scattering for lasso graph, J. Math. Phys., 54 (2013), 04210314, 14pp. doi: 10.1063/1.4799034.  Google Scholar [17] B. M. Levitan, Inverse Sturm-Liouville Problems, Nauka, Moscow, 1984 (Russian); English transl., VNU Sci. Press, Utrecht, 1987.  Google Scholar [18] V. A. Marchenko, Sturm-Liouville Operators and their Applications, Naukova Dumka, Kiev, 1977 (Russian); English transl., Birkhauser, 1986.  Google Scholar [19] V. Marchenko, K. Mochizuki and I. Trooshin, Inverse scattering on a graph, containing circle, Analytic Methods of Analysis and Differ, Equations: AMADE 2006, 237-243. Cambridge Sci. Publ., Cambridge, 2008.  Google Scholar [20] V. A. Marchenko and I. V. Ostrovskii, A characterization of the spectrum of the Hill operator, Mat. Sbornik, 97 (1975), 540-606 (Russian); English transl. in Math. USSR Sbornik, 26 (1975), 493-554.  Google Scholar [21] K. Mochizuki and I. Trooshin, On the scattering on a loop shaped graph, Evolution Equations of hyperbolic and Schroedinger Type, 227-245, Progr. Math., 301, Birkhauser/Springer. Basel A6, Basel, 2012. doi: 10.1007/978-3-0348-0454-7_12.  Google Scholar [22] V. N. Pivovarchik, Inverse problem for the Sturm-Liouville equation on a simple graph, SIAM J. Math. Anal., 32 (2000), 801-819.  doi: 10.1137/S0036141000368247.  Google Scholar [23] Yu. V. Pokorny, O. M. Penkin and V. L. Pryadiev, et al., Differential equations on geometrical graphs, Fizmatlit, Moscow, 2004 (Russian). Google Scholar [24] J. Pöschel and E. Trubowitz, Inverse Spectral Theory, New York, Academic Press, 1987.  Google Scholar [25] A. M. Savchuk, On the eigenvalues and eigenfunctions of the Sturm-Liouville operator with a singular potential, Mathematical Notes, 69 (2001), 245-252.  doi: 10.1023/A:1002880520696.  Google Scholar [26] I. V. Stankevich, An inverse problem of spectral analysis for Hill's equations, Doklady Akad. Nauk SSSR, 192 (1970), 34-37 (Russian).   Google Scholar [27] C.-F. Yang, Inverse spectral problems for the Sturm-Liouville operator on a $d$-star graph, J. Math. Anal. Appl., 365 (2010), 742-749.  doi: 10.1016/j.jmaa.2009.12.016.  Google Scholar [28] C.-F. Yang and X.-P. Yang, Uniqueness theorems from partial information of the potential on a graph, J. Inverse Ill-Posed Prob., 19 (2011), 631-641.  doi: 10.1515/JIIP.2011.059.  Google Scholar [29] V. A. Yurko, Inverse nodal problems for the Sturm-Liouville differential operators on a star-type graph, Siberian Math. J., 50 (2009), 373-378.  doi: 10.1007/s11202-009-0043-2.  Google Scholar [30] V. A. Yurko, Inverse problems for Sturm-Liouville operators on graphs with a cycle, Operators and Matrices, 2 (2008), 543-553.  doi: 10.7153/oam-02-34.  Google Scholar [31] V. A. Yurko, Inverse spectral problems for differential operators on spatial networks, Russian Mathematical Surveys, 71 (2016), 539-584.  doi: 10.4213/rm9709.  Google Scholar [32] V. A. Yurko, Inverse spectral problems for differential operators on arbitrary compact graphs, J. Inverse and Ill-Posed Probl., 18 (2010), 245-261.  doi: 10.1515/JIIP.2010.009.  Google Scholar
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