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 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. FreilingM. 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. FreilingM. 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.

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