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Lasso graph
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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
| 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 |
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 and 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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [7] |
G. Freiling and V. Yurko, Inverse Sturm-Liouville problems and their applications, Nova Science Publishers, Inc., Huntington, NY, 2001. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [14] |
P. Kuchment,
Quantum graphs. Some basic structures, Waves Random Media, 14 (2004), S107-S128.
doi: 10.1088/0959-7174/14/1/014. |
| [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. |
| [16] |
P. Kurasov, Inverse scattering for lasso graph, J. Math. Phys., 54 (2013), 04210314, 14pp.
doi: 10.1063/1.4799034. |
| [17] |
B. M. Levitan, Inverse Sturm-Liouville Problems, Nauka, Moscow, 1984 (Russian); English transl., VNU Sci. Press, Utrecht, 1987. |
| [18] |
V. A. Marchenko, Sturm-Liouville Operators and their Applications, Naukova Dumka, Kiev, 1977 (Russian); English transl., Birkhauser, 1986. |
| [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. |
| [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. |
| [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. |
| [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. |
| [23] |
Yu. V. Pokorny, O. M. Penkin and V. L. Pryadiev, et al., Differential equations on geometrical graphs, Fizmatlit, Moscow, 2004 (Russian). |
| [24] |
J. Pöschel and E. Trubowitz, Inverse Spectral Theory, New York, Academic Press, 1987. |
| [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. |
| [26] |
I. V. Stankevich,
An inverse problem of spectral analysis for Hill's equations, Doklady Akad. Nauk SSSR, 192 (1970), 34-37 (Russian).
|
| [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. |
| [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. |
| [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. |
| [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. |
| [31] |
V. A. Yurko,
Inverse spectral problems for differential operators on spatial networks, Russian Mathematical Surveys, 71 (2016), 539-584.
doi: 10.4213/rm9709. |
| [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. |
show all references
References:
| [1] |
G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, Amer. Math. Soc., Providence, RI, 2013. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [7] |
G. Freiling and V. Yurko, Inverse Sturm-Liouville problems and their applications, Nova Science Publishers, Inc., Huntington, NY, 2001. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [14] |
P. Kuchment,
Quantum graphs. Some basic structures, Waves Random Media, 14 (2004), S107-S128.
doi: 10.1088/0959-7174/14/1/014. |
| [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. |
| [16] |
P. Kurasov, Inverse scattering for lasso graph, J. Math. Phys., 54 (2013), 04210314, 14pp.
doi: 10.1063/1.4799034. |
| [17] |
B. M. Levitan, Inverse Sturm-Liouville Problems, Nauka, Moscow, 1984 (Russian); English transl., VNU Sci. Press, Utrecht, 1987. |
| [18] |
V. A. Marchenko, Sturm-Liouville Operators and their Applications, Naukova Dumka, Kiev, 1977 (Russian); English transl., Birkhauser, 1986. |
| [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. |
| [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. |
| [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. |
| [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. |
| [23] |
Yu. V. Pokorny, O. M. Penkin and V. L. Pryadiev, et al., Differential equations on geometrical graphs, Fizmatlit, Moscow, 2004 (Russian). |
| [24] |
J. Pöschel and E. Trubowitz, Inverse Spectral Theory, New York, Academic Press, 1987. |
| [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. |
| [26] |
I. V. Stankevich,
An inverse problem of spectral analysis for Hill's equations, Doklady Akad. Nauk SSSR, 192 (1970), 34-37 (Russian).
|
| [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. |
| [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. |
| [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. |
| [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. |
| [31] |
V. A. Yurko,
Inverse spectral problems for differential operators on spatial networks, Russian Mathematical Surveys, 71 (2016), 539-584.
doi: 10.4213/rm9709. |
| [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. |
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