American Institute of Mathematical Sciences

doi: 10.3934/ipi.2021025

A note on transmission eigenvalues in electromagnetic scattering theory

 1 Department of Mathematics, Rutgers University, Piscataway, NJ 08854-8019, USA 2 Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

* Corresponding author: jingni.xiao@rutgers.edu

Received  October 2020 Revised  January 2021 Published  March 2021

Fund Project: The author F. Cakoni is supported in part by the AFOSR Grant FA9550-20-1-0024 and NSF Grant DMS-1813492

This short note was motivated by our efforts to investigate whether there exists a half plane free of transmission eigenvalues for Maxwell's equations. This question is related to solvability of the time domain interior transmission problem which plays a fundamental role in the justification of linear sampling and factorization methods with time dependent data. Our original goal was to adapt semiclassical analysis techniques developed in [21,23] to prove that for some combination of electromagnetic parameters, the transmission eigenvalues lie in a strip around the real axis. Unfortunately we failed. To try to understand why, we looked at the particular example of spherically symmetric media, which provided us with some insight on why we couldn't prove the above result. Hence this paper reports our findings on the location of all transmission eigenvalues and the existence of complex transmission eigenvalues for Maxwell's equations for spherically stratified media. We hope that these results can provide reasonable conjectures for general electromagnetic media.

Citation: Fioralba Cakoni, Shixu Meng, Jingni Xiao. A note on transmission eigenvalues in electromagnetic scattering theory. Inverse Problems & Imaging, doi: 10.3934/ipi.2021025
References:
 [1] E. Blåsten, H. Liu and J. Xiao, On an electromagnetic problem in a corner and its applications, accepted in Anal. PDE, arXiv: 1901.00581. Google Scholar [2] F. Cakoni and D. Colton, A Qualitative Approach to Inverse Scattering Theory, vol. 188 of Applied Mathematical Sciences, Springer, New York, 2014. doi: 10.1007/978-1-4614-8827-9.  Google Scholar [3] F. Cakoni, D. Colton and H. Haddar, Inverse Scattering Theory and Transmission Eigenvalues, vol. 88 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2016. doi: 10.1137/1.9781611974461.  Google Scholar [4] F. Cakoni, D. Colton and P. Monk, The Linear Sampling Method in Inverse Electromagnetic Scattering, vol. 80 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2011. doi: 10.1137/1.9780898719406.  Google Scholar [5] F. Cakoni, D. Gintides and H. Haddar, The existence of an infinite discrete set of transmission eigenvalues, SIAM J. Math. Anal., 42 (2010), 237-255.  doi: 10.1137/090769338.  Google Scholar [6] F. Cakoni and A. Kirsch, On the interior transmission eigenvalue problem, Int. J. Comput. Sci. Math., 3 (2010), 142-167.  doi: 10.1504/IJCSM.2010.033932.  Google Scholar [7] F. Cakoni, P. Monk and V. Selgas, Analysis of the linear sampling method for imaging penetrable obstacles in the time domain, accepted in Anal. PDE. Google Scholar [8] F. Cakoni and H.-M. Nguyen, On the discreteness of transmission eigenvalues for the Maxwell's equations, SIAM J. Math. Anal., 53 (2021), 888-913.  doi: 10.1137/20M1335121.  Google Scholar [9] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, vol. 93 of Applied Mathematical Sciences, 4$^{th}$ edition, Springer, Cham, 2019. doi: 10.1007/978-3-030-30351-8.  Google Scholar [10] D. Colton and Y.-J. Leung, Complex eigenvalues and the inverse spectral problem for transmission eigenvalues, Inverse Problems, 29 (2013), 104008, 6 pp. doi: 10.1088/0266-5611/29/10/104008.  Google Scholar [11] D. Colton and Y.-J. Leung, The existence of complex transmission eigenvalues for spherically stratified media, Appl. Anal., 96 (2017), 39-47.  doi: 10.1080/00036811.2016.1210788.  Google Scholar [12] D. Colton, Y.-J. Leung and S. Meng, Distribution of complex transmission eigenvalues for spherically stratified media, Inverse Problems, 31 (2015), 035006, 19 pp. doi: 10.1088/0266-5611/31/3/035006.  Google Scholar [13] D. Colton and P. Monk, The inverse scattering problem for time-harmonic acoustic waves in a penetrable medium, Quart. J. Mech. Appl. Math., 40 (1987), 189-212.  doi: 10.1093/qjmam/40.2.189.  Google Scholar [14] H. Haddar and S. Meng, The spectral analysis of the interior transmission eigenvalue problem for Maxwell's equations, J. Math. Pures Appl., 120 (2018), 1-32.  doi: 10.1016/j.matpur.2018.10.004.  Google Scholar [15] M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta, The interior transmission problem and bounds on transmission eigenvalues, Math. Res. Lett., 18 (2011), 279-293.  doi: 10.4310/MRL.2011.v18.n2.a7.  Google Scholar [16] A. Kirsch, The denseness of the far field patterns for the transmission problem, IMA J. Appl. Math., 37 (1986), 213-225.  doi: 10.1093/imamat/37.3.213.  Google Scholar [17] A. Kirsch and F. Hettlich, The Mathematical Theory of Time-Harmonic Maxwell's equations, vol. 190 of Applied Mathematical Sciences, Springer, Cham, 2015. doi: 10.1007/978-3-319-11086-8.  Google Scholar [18] Y.-J. Leung and D. Colton, Complex transmission eigenvalues for spherically stratified media, Inverse Problems, 28 (2012), 075005, 9 pp. doi: 10.1088/0266-5611/28/7/075005.  Google Scholar [19] H.-M. Nguyen and Q.-H. Nguyen, Discreteness of interior transmission eigenvalues revisited, Calc. Var. Partial Differential Equations, 56 (2017), Paper No. 51, 38 pp. doi: 10.1007/s00526-017-1143-7.  Google Scholar [20] V. Petkov and G. Vodev, Localization of the interior transmission eigenvalues for a ball, Inverse Probl. Imaging, 11 (2017), 355-372.  doi: 10.3934/ipi.2017017.  Google Scholar [21] G. Vodev, Transmission eigenvalue-free regions, Comm. Math. Phys., 336 (2015), 1141-1166.  doi: 10.1007/s00220-015-2311-2.  Google Scholar [22] G. Vodev, Transmission eigenvalues for strictly concave domains, Math. Ann., 366 (2016), 301-336.  doi: 10.1007/s00208-015-1329-2.  Google Scholar [23] G. Vodev, High-frequency approximation of the interior Dirichlet-to-Neumann map and applications to the transmission eigenvalues, Anal. PDE, 11 (2018), 213-236.  doi: 10.2140/apde.2018.11.213.  Google Scholar

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References:
 [1] E. Blåsten, H. Liu and J. Xiao, On an electromagnetic problem in a corner and its applications, accepted in Anal. PDE, arXiv: 1901.00581. Google Scholar [2] F. Cakoni and D. Colton, A Qualitative Approach to Inverse Scattering Theory, vol. 188 of Applied Mathematical Sciences, Springer, New York, 2014. doi: 10.1007/978-1-4614-8827-9.  Google Scholar [3] F. Cakoni, D. Colton and H. Haddar, Inverse Scattering Theory and Transmission Eigenvalues, vol. 88 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2016. doi: 10.1137/1.9781611974461.  Google Scholar [4] F. Cakoni, D. Colton and P. Monk, The Linear Sampling Method in Inverse Electromagnetic Scattering, vol. 80 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2011. doi: 10.1137/1.9780898719406.  Google Scholar [5] F. Cakoni, D. Gintides and H. Haddar, The existence of an infinite discrete set of transmission eigenvalues, SIAM J. Math. Anal., 42 (2010), 237-255.  doi: 10.1137/090769338.  Google Scholar [6] F. Cakoni and A. Kirsch, On the interior transmission eigenvalue problem, Int. J. Comput. Sci. Math., 3 (2010), 142-167.  doi: 10.1504/IJCSM.2010.033932.  Google Scholar [7] F. Cakoni, P. Monk and V. Selgas, Analysis of the linear sampling method for imaging penetrable obstacles in the time domain, accepted in Anal. PDE. Google Scholar [8] F. Cakoni and H.-M. Nguyen, On the discreteness of transmission eigenvalues for the Maxwell's equations, SIAM J. Math. Anal., 53 (2021), 888-913.  doi: 10.1137/20M1335121.  Google Scholar [9] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, vol. 93 of Applied Mathematical Sciences, 4$^{th}$ edition, Springer, Cham, 2019. doi: 10.1007/978-3-030-30351-8.  Google Scholar [10] D. Colton and Y.-J. Leung, Complex eigenvalues and the inverse spectral problem for transmission eigenvalues, Inverse Problems, 29 (2013), 104008, 6 pp. doi: 10.1088/0266-5611/29/10/104008.  Google Scholar [11] D. Colton and Y.-J. Leung, The existence of complex transmission eigenvalues for spherically stratified media, Appl. Anal., 96 (2017), 39-47.  doi: 10.1080/00036811.2016.1210788.  Google Scholar [12] D. Colton, Y.-J. Leung and S. Meng, Distribution of complex transmission eigenvalues for spherically stratified media, Inverse Problems, 31 (2015), 035006, 19 pp. doi: 10.1088/0266-5611/31/3/035006.  Google Scholar [13] D. Colton and P. Monk, The inverse scattering problem for time-harmonic acoustic waves in a penetrable medium, Quart. J. Mech. Appl. Math., 40 (1987), 189-212.  doi: 10.1093/qjmam/40.2.189.  Google Scholar [14] H. Haddar and S. Meng, The spectral analysis of the interior transmission eigenvalue problem for Maxwell's equations, J. Math. Pures Appl., 120 (2018), 1-32.  doi: 10.1016/j.matpur.2018.10.004.  Google Scholar [15] M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta, The interior transmission problem and bounds on transmission eigenvalues, Math. Res. Lett., 18 (2011), 279-293.  doi: 10.4310/MRL.2011.v18.n2.a7.  Google Scholar [16] A. Kirsch, The denseness of the far field patterns for the transmission problem, IMA J. Appl. Math., 37 (1986), 213-225.  doi: 10.1093/imamat/37.3.213.  Google Scholar [17] A. Kirsch and F. Hettlich, The Mathematical Theory of Time-Harmonic Maxwell's equations, vol. 190 of Applied Mathematical Sciences, Springer, Cham, 2015. doi: 10.1007/978-3-319-11086-8.  Google Scholar [18] Y.-J. Leung and D. Colton, Complex transmission eigenvalues for spherically stratified media, Inverse Problems, 28 (2012), 075005, 9 pp. doi: 10.1088/0266-5611/28/7/075005.  Google Scholar [19] H.-M. Nguyen and Q.-H. Nguyen, Discreteness of interior transmission eigenvalues revisited, Calc. Var. Partial Differential Equations, 56 (2017), Paper No. 51, 38 pp. doi: 10.1007/s00526-017-1143-7.  Google Scholar [20] V. Petkov and G. Vodev, Localization of the interior transmission eigenvalues for a ball, Inverse Probl. Imaging, 11 (2017), 355-372.  doi: 10.3934/ipi.2017017.  Google Scholar [21] G. Vodev, Transmission eigenvalue-free regions, Comm. Math. Phys., 336 (2015), 1141-1166.  doi: 10.1007/s00220-015-2311-2.  Google Scholar [22] G. Vodev, Transmission eigenvalues for strictly concave domains, Math. Ann., 366 (2016), 301-336.  doi: 10.1007/s00208-015-1329-2.  Google Scholar [23] G. Vodev, High-frequency approximation of the interior Dirichlet-to-Neumann map and applications to the transmission eigenvalues, Anal. PDE, 11 (2018), 213-236.  doi: 10.2140/apde.2018.11.213.  Google Scholar
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