February  2020, 14(1): 153-169. doi: 10.3934/ipi.2019068

An inverse problem for the Sturm-Liouville pencil with arbitrary entire functions in the boundary condition

1. 

Department of Applied Mathematics, School of Science, Nanjing University of Science 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

4. 

School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

* Corresponding author: Natalia Bondarenko

Received  June 2019 Published  November 2019

The Sturm-Liouville pencil is studied with arbitrary entire functions of the spectral parameter, contained in one of the boundary conditions. We solve the inverse problem, that consists in recovering the pencil coefficients from a part of the spectrum satisfying some conditions. Our main results are 1) uniqueness, 2) constructive solution, 3) local solvability and stability of the inverse problem. Our method is based on the reduction to the Sturm-Liouville problem without the spectral parameter in the boundary conditions. We use a special vector-functional Riesz-basis for that reduction.

Citation: Chuan-Fu Yang, Natalia Pavlovna Bondarenko, Xiao-Chuan Xu. An inverse problem for the Sturm-Liouville pencil with arbitrary entire functions in the boundary condition. Inverse Problems & Imaging, 2020, 14 (1) : 153-169. doi: 10.3934/ipi.2019068
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P. J. Browne and B. D. Sleeman, A uniqueness theorem for inverse eigenparameter dependent Sturm-Liouville problems, Inverse Problems, 13 (1997), 1453-1462.  doi: 10.1088/0266-5611/13/6/003.  Google Scholar

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[26]

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V. A. Marchenko, Sturm-Liouville operators and their applications, Izdat. Naukova Dumka, Kiev, 1977,331pp.  Google Scholar

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O. Martinyuk and V. Pivovarchik, On the Hochstadt-Lieberman theorem, Inverse Problems, 26 (2010), 6pp.  doi: 10.1088/0266-5611/26/3/035011.  Google Scholar

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[31]

J. R. McLaughlinP. L. Polyakov and P. E. Sacks, Reconstruction of a spherically symmetric speed of sound, SIAM J. Appl. Math., 54 (1994), 1203-1223.  doi: 10.1137/S0036139992238218.  Google Scholar

[32]

J. R. McLaughlin, P. E. Sacks and M. Somasundaram, Inverse scattering in acoustic media using interior transmission eigenvalues, in Inverse Problems in Wave Propagation, IMA Vol. Math. Appl., 90, Springer, New York, 1997,357-374. doi: 10.1007/978-1-4612-1878-4_17.  Google Scholar

[33]

A. S. Ozkan, An impulsive Sturm-Liouville problem with boundary conditions containing Herglotz-Nevanlinna type functions, Appl. Math. Inf. Sci., 9 (2015), 205-211.   Google Scholar

[34]

V. Pivovarchik, On the Hald-Gesztesy-Simon theorem, Integral Equations Operator Theory, 73 (2012), 383-393.  doi: 10.1007/s00020-012-1966-8.  Google Scholar

[35]

J. Pöschel and E. Trubowitz, Inverse Spectral Theory, Pure and Applied Mathematics, 130, Academic Press, Inc. Boston, MA, 1987.  Google Scholar

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L. Sakhnovich, Half-inverse problem on the finite interval, Inverse Problems, 17 (2001), 527-532.  doi: 10.1088/0266-5611/17/3/311.  Google Scholar

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A. M. Sedletskii, Nonharmonic analysis, J. Math. Sci. (N. Y.), 116 (2003), 3551-3619.  doi: 10.1023/A:1024107924340.  Google Scholar

[38]

C.-T. Shieh and V. A. Yurko, Inverse nodal and inverse spectral problems for discontinuous boundary value problems, J. Math. Anal. Appl., 347 (2008), 266-272.  doi: 10.1016/j.jmaa.2008.05.097.  Google Scholar

[39]

X.-C. XuC.-F. YangS. A. Buterin and V. A. Yurko, Estimates of complex eigenvalues and an inverse spectral problem for the transmission eigenvalue problem, Electron. J. Qual. Theory Differ. Equ., (2019), 15pp.  doi: 10.14232/ejqtde.2019.1.38.  Google Scholar

[40]

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

[41]

C.-F. Yang and N. P. Bondarenko, Reconstruction and solvability for discontinuous Hochstadt-Lieberman problems, preprint, arXiv: 1904.10263. Google Scholar

[42]

C.-F. Yang and N. P. Bondarenko, Local solvability and stability of inverse problems for Sturm-Liouville operators with discontinuity, preprint, arXiv: 1906.06552. Google Scholar

[43]

C.-F. Yang and N. P. Bondarenko, A partial inverse problem for the Sturm-Liouville operator on the Lasso-graph, Inverse Probl. Imaging, 13 (2019), 69-79.  doi: 10.3934/ipi.2019004.  Google Scholar

[44]

C.-F. Yang and Z.-Y. Huang, A half-inverse problem with eigenparameter dependent boundary conditions, Numer. Funct. Anal. Optim., 31 (2010), 754-762.  doi: 10.1080/01630563.2010.490934.  Google Scholar

[45]

C.-F. Yang and X.-C. Xu, Ambarzumyan-type theorem with polynomially dependent eigenparameter, Math. Methods Appl. Sci., 38 (2015), 4411-4415.  doi: 10.1002/mma.3380.  Google Scholar

[46]

V. A. Yurko, An inverse problem for pencils of differential operators, Mat. Sb., 191 (2000), 137-160.  doi: 10.1070/SM2000v191n10ABEH000520.  Google Scholar

[47]

V. A. Yurko, Inverse spectral problems for differential operators on spatial networks, Uspekhi Mat. Nauk, 71 (2016), 149-196.  doi: 10.4213/rm9709.  Google Scholar

show all references

References:
[1]

N. K. Bari, Biorthogonal systems and bases in Hilbert space, Moskov. Gos. Univ. Učenye Zapiski Matematika, 148 (1951), 69-107.   Google Scholar

[2]

G. Berkolaiko, R. Carlson, S. Fulling and P. Kuchment, Quantum Graphs and Their Applications, Contemporary Mathematics, 415, American Mathematical Society, Providence, RI, 2006. doi: 10.1090/conm/415.  Google Scholar

[3]

P. A. BindingP. J. Browne and B. A. Watson, Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter, J. Comput. Appl. Math., 148 (2002), 147-168.  doi: 10.1016/S0377-0427(02)00579-4.  Google Scholar

[4]

P. A. BindingP. J. Browne and B. A. Watson, Equivalence of inverse Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter, J. Math. Anal. Appl., 291 (2004), 246-261.  doi: 10.1016/j.jmaa.2003.11.025.  Google Scholar

[5]

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

[6]

N. Bondarenko and S. Buterin, On a local solvability and stability of the inverse transmission eigenvalue problem, Inverse Problems, 33 (2017), 19pp.  doi: 10.1088/1361-6420/aa8cb5.  Google Scholar

[7]

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

[8]

N. P. Bondarenko, Inverse problem for the differential pencil on an arbitrary graph with partial information given on the coefficients, Anal. Math. Phys., 9 (2019), 1393-1409.  doi: 10.1007/s13324-018-0244-6.  Google Scholar

[9]

P. J. Browne and B. D. Sleeman, A uniqueness theorem for inverse eigenparameter dependent Sturm-Liouville problems, Inverse Problems, 13 (1997), 1453-1462.  doi: 10.1088/0266-5611/13/6/003.  Google Scholar

[10]

S. A. Buterin, On half inverse problem for differential pencils with the spectral parameter in boundary conditions, Tamkang J. Math., 42 (2011), 355-364.  doi: 10.5556/j.tkjm.42.2011.912.  Google Scholar

[11]

S. Buterin and M. Kuznetsova, On Borg's method for non-selfadjoint Sturm-Liouville operators, Anal. Math. Phys., (2019), 1-18.  doi: 10.1007/s13324-019-00307-9.  Google Scholar

[12]

F. CakoniD. Colton and P. Monk, On the use of transmission eigenvalues to estimate the index of refraction from far field data, Inverse Problems, 23 (2007), 507-522.  doi: 10.1088/0266-5611/23/2/004.  Google Scholar

[13]

A. Y. Chernozhukova and G. Freiling, A uniqueness theorem for inverse spectral problems depending nonlinearly on the spectral parameter, Inverse Probl. Sci. Eng., 17 (2009), 777-785.  doi: 10.1080/17415970802538550.  Google Scholar

[14]

O. Christensen, An Introduction to Frames and Riesz Bases, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2003. doi: 10.1007/978-0-8176-8224-8.  Google Scholar

[15]

M. V. Chugunova, Inverse spectral problem for the Sturm-Liouville operator with eigenvalue parameter dependent boundary conditions, in Operator Theory, System Theory and Related Topics, Oper. Theory Adv. Appl., 123, Birkhäuser, Basel, 2001,187-194. doi: 10.1007/978-3-0348-8247-7_8.  Google Scholar

[16]

P. Exner, J. P. Keating, P. Kuchment, T. Sunada and A. Teplyaev, Analysis on Graphs and Its Applications, Proc. Sympos. Pure Math., 77, Amer. Math. Soc., Providence, RI, 2008. doi: 10.1090/pspum/077.  Google Scholar

[17]

G. Freiling and V. Yurko, Inverse Sturm-Liouville Problems and Their Applications, Nova Science Publishers, Inc., Huntington, NY, 2001.  Google Scholar

[18]

G. Freiling and V. A. Yurko, Inverse problems for Sturm-Liouville equations with boundary conditions polynomially dependent on the spectral parameter, Inverse Problems, 26 (2010), 17pp.  doi: 10.1088/0266-5611/26/5/055003.  Google Scholar

[19]

G. Freiling and V. Yurko, Determination of singular differential pencils from the Weyl function, Adv. Dyn. Syst. Appl., 7 (2012), 171-193.   Google Scholar

[20]

F. Gesztesy and B. Simon, Inverse spectral analysis with partial information on the potential. Ⅱ: The case of discrete spectrum, Trans. Amer. Math. Soc., 352 (2000), 2765-2787.  doi: 10.1090/S0002-9947-99-02544-1.  Google Scholar

[21]

N. J. Guliyev, Inverse eigenvalue problems for Sturm-Liouville equations with spectral parameter linearly contained in one of the boundary condition, Inverse Problems, 21 (2005), 1315-1330.  doi: 10.1088/0266-5611/21/4/008.  Google Scholar

[22]

N. J. Guliyev, Schrödinger operators with distributional potentials and boundary conditions dependent on the eigenvalue parameter, J. Math. Phys., 60 (2019), 23pp.  doi: 10.1063/1.5048692.  Google Scholar

[23]

O. Hald, Discontinuous inverse eigenvalue problem, Commun. Pure Appl. Math., 37 (1984), 539-577.  doi: 10.1002/cpa.3160370502.  Google Scholar

[24]

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

[25]

M. Horváth, On the inverse spectral theory of Schrödinger and Dirac operators, Trans. Amer. Math. Soc., 353 (2001), 4155-4171.  doi: 10.1090/S0002-9947-01-02765-9.  Google Scholar

[26]

O. R. Hryniv and Y. 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

[27]

B. M. Levitan, Inverse Sturm-Liouville problems, Nauka, Moscow, 1984,240pp.  Google Scholar

[28]

V. A. Marchenko, Sturm-Liouville operators and their applications, Izdat. Naukova Dumka, Kiev, 1977,331pp.  Google Scholar

[29]

O. Martinyuk and V. Pivovarchik, On the Hochstadt-Lieberman theorem, Inverse Problems, 26 (2010), 6pp.  doi: 10.1088/0266-5611/26/3/035011.  Google Scholar

[30]

J. R. McLaughlin and P. L. Polyakov, On the uniqueness of a spherically symmetric speed of sound from transmission eigenvalues, J. Differential Equations, 107 (1994), 351-382.  doi: 10.1006/jdeq.1994.1017.  Google Scholar

[31]

J. R. McLaughlinP. L. Polyakov and P. E. Sacks, Reconstruction of a spherically symmetric speed of sound, SIAM J. Appl. Math., 54 (1994), 1203-1223.  doi: 10.1137/S0036139992238218.  Google Scholar

[32]

J. R. McLaughlin, P. E. Sacks and M. Somasundaram, Inverse scattering in acoustic media using interior transmission eigenvalues, in Inverse Problems in Wave Propagation, IMA Vol. Math. Appl., 90, Springer, New York, 1997,357-374. doi: 10.1007/978-1-4612-1878-4_17.  Google Scholar

[33]

A. S. Ozkan, An impulsive Sturm-Liouville problem with boundary conditions containing Herglotz-Nevanlinna type functions, Appl. Math. Inf. Sci., 9 (2015), 205-211.   Google Scholar

[34]

V. Pivovarchik, On the Hald-Gesztesy-Simon theorem, Integral Equations Operator Theory, 73 (2012), 383-393.  doi: 10.1007/s00020-012-1966-8.  Google Scholar

[35]

J. Pöschel and E. Trubowitz, Inverse Spectral Theory, Pure and Applied Mathematics, 130, Academic Press, Inc. Boston, MA, 1987.  Google Scholar

[36]

L. Sakhnovich, Half-inverse problem on the finite interval, Inverse Problems, 17 (2001), 527-532.  doi: 10.1088/0266-5611/17/3/311.  Google Scholar

[37]

A. M. Sedletskii, Nonharmonic analysis, J. Math. Sci. (N. Y.), 116 (2003), 3551-3619.  doi: 10.1023/A:1024107924340.  Google Scholar

[38]

C.-T. Shieh and V. A. Yurko, Inverse nodal and inverse spectral problems for discontinuous boundary value problems, J. Math. Anal. Appl., 347 (2008), 266-272.  doi: 10.1016/j.jmaa.2008.05.097.  Google Scholar

[39]

X.-C. XuC.-F. YangS. A. Buterin and V. A. Yurko, Estimates of complex eigenvalues and an inverse spectral problem for the transmission eigenvalue problem, Electron. J. Qual. Theory Differ. Equ., (2019), 15pp.  doi: 10.14232/ejqtde.2019.1.38.  Google Scholar

[40]

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

[41]

C.-F. Yang and N. P. Bondarenko, Reconstruction and solvability for discontinuous Hochstadt-Lieberman problems, preprint, arXiv: 1904.10263. Google Scholar

[42]

C.-F. Yang and N. P. Bondarenko, Local solvability and stability of inverse problems for Sturm-Liouville operators with discontinuity, preprint, arXiv: 1906.06552. Google Scholar

[43]

C.-F. Yang and N. P. Bondarenko, A partial inverse problem for the Sturm-Liouville operator on the Lasso-graph, Inverse Probl. Imaging, 13 (2019), 69-79.  doi: 10.3934/ipi.2019004.  Google Scholar

[44]

C.-F. Yang and Z.-Y. Huang, A half-inverse problem with eigenparameter dependent boundary conditions, Numer. Funct. Anal. Optim., 31 (2010), 754-762.  doi: 10.1080/01630563.2010.490934.  Google Scholar

[45]

C.-F. Yang and X.-C. Xu, Ambarzumyan-type theorem with polynomially dependent eigenparameter, Math. Methods Appl. Sci., 38 (2015), 4411-4415.  doi: 10.1002/mma.3380.  Google Scholar

[46]

V. A. Yurko, An inverse problem for pencils of differential operators, Mat. Sb., 191 (2000), 137-160.  doi: 10.1070/SM2000v191n10ABEH000520.  Google Scholar

[47]

V. A. Yurko, Inverse spectral problems for differential operators on spatial networks, Uspekhi Mat. Nauk, 71 (2016), 149-196.  doi: 10.4213/rm9709.  Google Scholar

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