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 and Imaging, 2020, 14 (1) : 153-169. doi: 10.3934/ipi.2019068
References:
[1]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

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

[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.

[19]

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

[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.

[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.

[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.

[23]

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

[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.

[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.

[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.

[27]

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

[28]

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

[29]

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

[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.

[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.

[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.

[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. 

[34]

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

[35]

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

[36]

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

[37]

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

[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.

[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.

[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.

[41]

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

[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.

[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.

[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.

[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.

[46]

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

[47]

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

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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

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

[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.

[19]

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

[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.

[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.

[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.

[23]

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

[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.

[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.

[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.

[27]

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

[28]

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

[29]

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

[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.

[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.

[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.

[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. 

[34]

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

[35]

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

[36]

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

[37]

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

[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.

[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.

[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.

[41]

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

[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.

[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.

[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.

[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.

[46]

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

[47]

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

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