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. 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.  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. 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.  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
Figure 2.  Plots for equation (7), $m = 5$
[1]

Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463

[2]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[3]

Andy Hammerlindl, Jana Rodriguez Hertz, Raúl Ures. Ergodicity and partial hyperbolicity on Seifert manifolds. Journal of Modern Dynamics, 2020, 16: 331-348. doi: 10.3934/jmd.2020012

[4]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451

[5]

Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073

[6]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[7]

Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020076

[8]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345

[9]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047

[10]

Yi-Hsuan Lin, Gen Nakamura, Roland Potthast, Haibing Wang. Duality between range and no-response tests and its application for inverse problems. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020072

[11]

Kha Van Huynh, Barbara Kaltenbacher. Some application examples of minimization based formulations of inverse problems and their regularization. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020074

[12]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[13]

Shiqiu Fu, Kanishka Perera. On a class of semipositone problems with singular Trudinger-Moser nonlinearities. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020452

[14]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376

[15]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440

[16]

Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251

[17]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253

[18]

Wenbin Li, Jianliang Qian. Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020073

[19]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

[20]

Anna Canale, Francesco Pappalardo, Ciro Tarantino. Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020274

2019 Impact Factor: 1.373

Metrics

  • PDF downloads (125)
  • HTML views (256)
  • Cited by (1)

Other articles
by authors

[Back to Top]