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doi: 10.3934/ipi.2021073
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A new approach to the inverse discrete transmission eigenvalue problem

1. 

Department of Applied Mathematics and Physics, Samara National Research University, Moskovskoye Shosse 34, Samara 443086, Russia

2. 

Department of Mechanics and Mathematics, Saratov State University, Astrakhanskaya 83, Saratov 410012, Russia

*Corresponding author: Natalia P. Bondarenko

Received  June 2021 Revised  September 2021 Early access December 2021

Fund Project: The authors are supported by RFBR grants 20-31-70005, 19-01-00102

A discrete analog is considered for the inverse transmission eigenvalue problem, having applications in acoustics. We provide a well-posed inverse problem statement, develop a constructive procedure for solving this problem, prove uniqueness of solution, global solvability, local solvability, and stability. Our approach is based on the reduction of the discrete transmission eigenvalue problem to a linear system with polynomials of the spectral parameter in the boundary condition.

Citation: Natalia P. Bondarenko, Vjacheslav A. Yurko. A new approach to the inverse discrete transmission eigenvalue problem. Inverse Problems and Imaging, doi: 10.3934/ipi.2021073
References:
[1]

T. AktosunD. Gintides and V. G. Papanicolaou, The uniqueness in the inverse problem for transmission eigenvalues for the spherically symmetric variable-speed wave equation, Inverse Problems, 27 (2011), 115004.  doi: 10.1088/0266-5611/27/11/115004.

[2]

T. Aktosun and V. G. Papanicolaou, Inverse problem with transmission eigenvalues for the discrete Schrödinger equation, J. Math. Phys., 56 (2015), 082101.  doi: 10.1063/1.4927264.

[3]

N. P. Bondarenko, Inverse Sturm-Liouville problem with analytical functions in the boundary condition, Open Math., 18 (2020), 512-528.  doi: 10.1515/math-2020-0188.

[4]

N. P. Bondarenko, Solvability and stability of the inverse Sturm-Liouville problem with analytical functions in the boundary condition, Math. Meth. Appl. Sci., 43 (2020), 7009-7021.  doi: 10.1002/mma.6451.

[5]

N. P. Bondarenko, A partial inverse Sturm-Liouville problem on an arbitrary graph, Math. Meth. Appl. Sci., 44 (2021), 6896-6910.  doi: 10.1002/mma.7231.

[6]

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.

[7]

N. P. Bondarenko and S. Buterin, On a local solvability and stability of the inverse transmission eigenvalue problem, Inverse Problems, 33 (2017), 115010.  doi: 10.1088/1361-6420/aa8cb5.

[8]

S. A. ButerinA. E. Choque-Rivero and M. A. Kuznetsova, On a regularization approach to the inverse transmission eigenvalue problem, Inverse Problems, 36 (2020), 105002.  doi: 10.1088/1361-6420/abaf3c.

[9]

S. A. Buterin and C.-F. Yang, On an inverse transmission problem from complex eigenvalues, Results Math., 71 (2017), 859-866.  doi: 10.1007/s00025-015-0512-9.

[10]

S. A. ButerinC.-F. Yang and V. A. Yurko, On an open question in the inverse transmission eigenvalue problem, Inverse Problems, 31 (2015), 045003.  doi: 10.1088/0266-5611/31/4/045003.

[11]

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.

[12]

D. Colton and Y.-J. Leung, Complex eigenvalues and the inverse spectral problem for transmission eigenvalues, Inverse Problems, 29 (2013), 104008.  doi: 10.1088/0266-5611/29/10/104008.

[13]

D. Gintides and N. Pallikarakis, The inverse transmission eigenvalue problem for a discontinuous refractive index, Inverse Problems, 33 (2017), 055006.  doi: 10.1088/1361-6420/aa5bf0.

[14]

H. Hochstadt, On the construction of a Jacobi matrix from spectral data, Linear Algebra Appl., 8 (1974), 435-446.  doi: 10.1016/0024-3795(74)90077-9.

[15]

H. Hochstadt, On the construction of a Jacobi matrix from mixed given data, Lin. Alg. Appl., 28 (1979), 113-115.  doi: 10.1016/0024-3795(79)90124-1.

[16]

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.

[17]

R. O. 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.

[18]

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

[19]

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

[20]

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.

[21]

V. G. Papanicolaou and A. V. Doumas, On the discrete one-dimensional inverse transmission eigenvalue problem, Inverse Problems, 27 (2011), 015004.  doi: 10.1088/0266-5611/27/1/015004.

[22]

G. Wei, The uniqueness for inverse discrete transmission eigenvalue problems, Linear Algebra Appl., 439 (2013), 3699-3712.  doi: 10.1016/j.laa.2013.10.027.

[23]

Z. Wei and G. Wei, The inverse discrete transmission eigenvalue problem for absorbing media, Inverse Probl. Sci. Eng., 26 (2018), 83-99.  doi: 10.1080/17415977.2017.1309397.

[24]

X.-C. Xu and C.-F. Yang, On the inverse spectral stability for the transmission eigenvalue problem with finite data, Inverse Problems, 36 (2020), 085006.  doi: 10.1088/1361-6420/ab9590.

[25]

V. A. Yurko, An inverse problem for operators of a triangular structure, Results Math., 30 (1996), 346-373.  doi: 10.1007/BF03322200.

[26]

V. A. Yurko, Inverse Spectral Problems for Differential Operators and their Applications, Analytical Methods and Special Functions, Gordon and Breach Science Publishers, Amsterdam, 2000.

show all references

References:
[1]

T. AktosunD. Gintides and V. G. Papanicolaou, The uniqueness in the inverse problem for transmission eigenvalues for the spherically symmetric variable-speed wave equation, Inverse Problems, 27 (2011), 115004.  doi: 10.1088/0266-5611/27/11/115004.

[2]

T. Aktosun and V. G. Papanicolaou, Inverse problem with transmission eigenvalues for the discrete Schrödinger equation, J. Math. Phys., 56 (2015), 082101.  doi: 10.1063/1.4927264.

[3]

N. P. Bondarenko, Inverse Sturm-Liouville problem with analytical functions in the boundary condition, Open Math., 18 (2020), 512-528.  doi: 10.1515/math-2020-0188.

[4]

N. P. Bondarenko, Solvability and stability of the inverse Sturm-Liouville problem with analytical functions in the boundary condition, Math. Meth. Appl. Sci., 43 (2020), 7009-7021.  doi: 10.1002/mma.6451.

[5]

N. P. Bondarenko, A partial inverse Sturm-Liouville problem on an arbitrary graph, Math. Meth. Appl. Sci., 44 (2021), 6896-6910.  doi: 10.1002/mma.7231.

[6]

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.

[7]

N. P. Bondarenko and S. Buterin, On a local solvability and stability of the inverse transmission eigenvalue problem, Inverse Problems, 33 (2017), 115010.  doi: 10.1088/1361-6420/aa8cb5.

[8]

S. A. ButerinA. E. Choque-Rivero and M. A. Kuznetsova, On a regularization approach to the inverse transmission eigenvalue problem, Inverse Problems, 36 (2020), 105002.  doi: 10.1088/1361-6420/abaf3c.

[9]

S. A. Buterin and C.-F. Yang, On an inverse transmission problem from complex eigenvalues, Results Math., 71 (2017), 859-866.  doi: 10.1007/s00025-015-0512-9.

[10]

S. A. ButerinC.-F. Yang and V. A. Yurko, On an open question in the inverse transmission eigenvalue problem, Inverse Problems, 31 (2015), 045003.  doi: 10.1088/0266-5611/31/4/045003.

[11]

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.

[12]

D. Colton and Y.-J. Leung, Complex eigenvalues and the inverse spectral problem for transmission eigenvalues, Inverse Problems, 29 (2013), 104008.  doi: 10.1088/0266-5611/29/10/104008.

[13]

D. Gintides and N. Pallikarakis, The inverse transmission eigenvalue problem for a discontinuous refractive index, Inverse Problems, 33 (2017), 055006.  doi: 10.1088/1361-6420/aa5bf0.

[14]

H. Hochstadt, On the construction of a Jacobi matrix from spectral data, Linear Algebra Appl., 8 (1974), 435-446.  doi: 10.1016/0024-3795(74)90077-9.

[15]

H. Hochstadt, On the construction of a Jacobi matrix from mixed given data, Lin. Alg. Appl., 28 (1979), 113-115.  doi: 10.1016/0024-3795(79)90124-1.

[16]

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.

[17]

R. O. 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.

[18]

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

[19]

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

[20]

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.

[21]

V. G. Papanicolaou and A. V. Doumas, On the discrete one-dimensional inverse transmission eigenvalue problem, Inverse Problems, 27 (2011), 015004.  doi: 10.1088/0266-5611/27/1/015004.

[22]

G. Wei, The uniqueness for inverse discrete transmission eigenvalue problems, Linear Algebra Appl., 439 (2013), 3699-3712.  doi: 10.1016/j.laa.2013.10.027.

[23]

Z. Wei and G. Wei, The inverse discrete transmission eigenvalue problem for absorbing media, Inverse Probl. Sci. Eng., 26 (2018), 83-99.  doi: 10.1080/17415977.2017.1309397.

[24]

X.-C. Xu and C.-F. Yang, On the inverse spectral stability for the transmission eigenvalue problem with finite data, Inverse Problems, 36 (2020), 085006.  doi: 10.1088/1361-6420/ab9590.

[25]

V. A. Yurko, An inverse problem for operators of a triangular structure, Results Math., 30 (1996), 346-373.  doi: 10.1007/BF03322200.

[26]

V. A. Yurko, Inverse Spectral Problems for Differential Operators and their Applications, Analytical Methods and Special Functions, Gordon and Breach Science Publishers, Amsterdam, 2000.

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