Inverse transmission eigenvalue problem for fixed angular momentum

Xin-Jian Xu; Chuan-Fu Yang; Xiao-Chuan Xu

doi:10.3934/ipi.2022039

Article Contents

2023, Volume 17, Issue 1: 263-274. Doi: 10.3934/ipi.2022039

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Inverse transmission eigenvalue problem for fixed angular momentum

1.
Department of Mathematics, School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, China
2.
School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, Jiangsu, China

^*Corresponding authors
^*Corresponding authors

Received: March 2022

Revised: June 2022

Early access: July 27, 2022

Published online: July 27, 2022

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Abstract

We consider the inverse problem of the interior transmission eigenvalue problem for spherically stratified media with fixed angular-momentum quantum number $ l $. Under the case $ \int^1_0\sqrt{\rho(r)}{\rm d}r\leq1 $, we show that a knowledge of all the transmission eigenvalues for $ l $ uniquely determine the refraction index $ \rho(r) $ on $ [0, 1] $. And under the case $ \int^1_0\sqrt{\rho(r)}{\rm d}r>1 $, we show that all the transmission eigenvalues for fixed $ l $ with a partial information of the refraction index $ \rho(r) $ imply the uniqueness of $ \rho(r) $ on $ [0, 1] $.

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Mathematics Subject Classification: 34A55, 34B24, 47E05.

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References

[1]	M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions, New York: Dover, 1972.
[2]	T. Aktosun, D. 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.
[3]	N. 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.
[4]	S. A. Buterin, A. 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.
[5]	S. Buterin, C-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.
[6]	F. Cakoni, D. Colton and D. Gintides, The interior transmission eigenvalue problem, SIAM Journal on Mathematical Analysis, 42 (2010), 2912-2921. doi: 10.1137/100793542.
[7]	F. Cakoni, D. Gintides and H. Haddar, The existence of an infinite discrete set of transmission eigenvalues, SIAM Journal on Mathematical Analysis, 42 (2010), 237-255. doi: 10.1137/090769338.
[8]	L.-H. Chen, An inverse uniqueness on the index of refraction with a transition region, Applicable Analysis, 95 (2016), 545-561. doi: 10.1080/00036811.2015.1022154.
[9]	L.-H. Chen, A fixed energy fixed angle inverse scattering in interior transmission problem, Reports on Mathematical Physics, 79 (2017), 331-345. doi: 10.1016/S0034-4877(17)30047-2.
[10]	D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3$^{nd}$ edition, New York: Springer, 2013. doi: 10.1007/978-1-4614-4942-3.
[11]	D. Colton and Yuk-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.
[12]	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.
[13]	A. Kostenko, A. Sakhnovich and G. Teschl, Inverse eigenvalue problems for perturbed spherical Schrödinger operators, Inverse Problems, 26 (2010), 105013. doi: 10.1088/0266-5611/26/10/105013.
[14]	J. R. McLaughlin and P. L. Polyakov, On the uniqueness of a spherically symmetric speed of sound from transmission eigenvalues, Journal of Differential Equations, 107 (1994), 351-382. doi: 10.1006/jdeq.1994.1017.
[15]	L. S. Pontryagin, Ordinary Differential Equations, Translated from the Russian, Addison-Wesley, 1962.
[16]	Y. P. Wang and C. T. Shieh, The inverse interior transmission eigenvalue problem with mixed spectral data, Applied Mathematics and Computation, 343 (2019), 285-298. doi: 10.1016/j.amc.2018.09.014.
[17]	Z. Wei and G. Wei, Unique reconstruction of the potential for the interior transmission eigenvalue problem for spherically stratified media, Inverse Problems, 36 (2020), 035017. doi: 10.1088/1361-6420/ab6e77.
[18]	G. Wei and H.-K. Xu, Inverse spectral analysis for the transmission eigenvalue problem, Inverse Problems, 29 (2013), 115012. doi: 10.1088/0266-5611/29/11/115012.
[19]	X.-C. Xu, L.-J. Ma and C.-F. Yang, On the stability of the inverse transmission eigenvalue problem from the data of McLaughlin and Polyakov, Journal of Differential Equations, 316 (2022), 222-248. doi: 10.1016/j.jde.2022.01.052.
[20]	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.
[21]	X.-C. Xu, C.-F. Yang, S. Buterin and V. Yurko, Estimates of complex eigenvalues and an inverse spectral problem for the transmission eigenvalue problem, Electronic Journal of Qualitative Theory of Differential Equations, 38 (2019), 1-15. doi: 10.14232/ejqtde.2019.1.38.
[22]	B. Ya Levin, Lectures on Entire Functions, AMS Translations, Providence, RI, 1996. doi: 10.1090/mmono/150.
[23]	C.-F. Yang and S. Buterin, Uniqueness of the interior transmission problem with partial information on the potential and eigenvalues, Journal of Differential Equations, 260 (2016), 4871-4887. doi: 10.1016/j.jde.2015.11.031.
[24]	C.-F. Yang and S. Buterin, Isospectral sets for transmission eigenvalue problem, Journal of Inverse and Ill-Posed Problems, 28 (2020), 63-69. doi: 10.1515/jiip-2018-0058.