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

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

Chuan-Fu Yang 1, and  Natalia Pavlovna Bondarenko 2,3,

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.

Keywords: Partial inverse spectral problem, Sturm-Liouville operator on a lasso graph, quantum graph, singular potential.
Mathematics Subject Classification: 34A55, 34B05, 34B09, 34B45, 34L20, 34L40, 47E05.
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

show all references

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

Figure 1.  Lasso graph
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Figure 2.  Plots for equation (7), $m = 5$
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