-
Previous Article
Two-dimensional inverse scattering for quasi-linear biharmonic operator
- IPI Home
- This Issue
-
Next Article
Near-field imaging for an obstacle above rough surfaces with limited aperture data
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 |
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 [
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[21] |
G. Vodev,
Transmission eigenvalue-free regions, Comm. Math. Phys., 336 (2015), 1141-1166.
doi: 10.1007/s00220-015-2311-2. |
[22] |
G. Vodev,
Transmission eigenvalues for strictly concave domains, Math. Ann., 366 (2016), 301-336.
doi: 10.1007/s00208-015-1329-2. |
[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. |
show all references
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[21] |
G. Vodev,
Transmission eigenvalue-free regions, Comm. Math. Phys., 336 (2015), 1141-1166.
doi: 10.1007/s00220-015-2311-2. |
[22] |
G. Vodev,
Transmission eigenvalues for strictly concave domains, Math. Ann., 366 (2016), 301-336.
doi: 10.1007/s00208-015-1329-2. |
[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. |
[1] |
Fioralba Cakoni, Drossos Gintides. New results on transmission eigenvalues. Inverse Problems and Imaging, 2010, 4 (1) : 39-48. doi: 10.3934/ipi.2010.4.39 |
[2] |
Andreas Kirsch. On the existence of transmission eigenvalues. Inverse Problems and Imaging, 2009, 3 (2) : 155-172. doi: 10.3934/ipi.2009.3.155 |
[3] |
Johannes Elschner, Guanghui Hu. Uniqueness in inverse transmission scattering problems for multilayered obstacles. Inverse Problems and Imaging, 2011, 5 (4) : 793-813. doi: 10.3934/ipi.2011.5.793 |
[4] |
Vesselin Petkov, Georgi Vodev. Localization of the interior transmission eigenvalues for a ball. Inverse Problems and Imaging, 2017, 11 (2) : 355-372. doi: 10.3934/ipi.2017017 |
[5] |
Armin Lechleiter. The factorization method is independent of transmission eigenvalues. Inverse Problems and Imaging, 2009, 3 (1) : 123-138. doi: 10.3934/ipi.2009.3.123 |
[6] |
Luc Robbiano. Counting function for interior transmission eigenvalues. Mathematical Control and Related Fields, 2016, 6 (1) : 167-183. doi: 10.3934/mcrf.2016.6.167 |
[7] |
Zhiyuan Wen, Meirong Zhang. On the optimization problems of the principal eigenvalues of measure differential equations with indefinite measures. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3257-3274. doi: 10.3934/dcdsb.2020061 |
[8] |
Yalin Zhang, Guoliang Shi. Continuous dependence of the transmission eigenvalues in one dimension. Inverse Problems and Imaging, 2015, 9 (1) : 273-287. doi: 10.3934/ipi.2015.9.273 |
[9] |
Ha Pham, Plamen Stefanov. Weyl asymptotics of the transmission eigenvalues for a constant index of refraction. Inverse Problems and Imaging, 2014, 8 (3) : 795-810. doi: 10.3934/ipi.2014.8.795 |
[10] |
Fioralba Cakoni, Anne Cossonnière, Houssem Haddar. Transmission eigenvalues for inhomogeneous media containing obstacles. Inverse Problems and Imaging, 2012, 6 (3) : 373-398. doi: 10.3934/ipi.2012.6.373 |
[11] |
Abdessatar Khelifi, Siwar Saidani. Asymptotic behavior of eigenvalues of the Maxwell system in the presence of small changes in the interface of an inclusion. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022080 |
[12] |
Fioralba Cakoni, Shari Moskow, Scott Rome. The perturbation of transmission eigenvalues for inhomogeneous media in the presence of small penetrable inclusions. Inverse Problems and Imaging, 2015, 9 (3) : 725-748. doi: 10.3934/ipi.2015.9.725 |
[13] |
Jun Zhang, Xinyue Fan. An efficient spectral method for the Helmholtz transmission eigenvalues in polar geometries. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4799-4813. doi: 10.3934/dcdsb.2019031 |
[14] |
Fioralba Cakoni, Shari Moskow, Scott Rome. Asymptotic expansions of transmission eigenvalues for small perturbations of media with generally signed contrast. Inverse Problems and Imaging, 2018, 12 (4) : 971-992. doi: 10.3934/ipi.2018041 |
[15] |
Julián Fernández Bonder, Analía Silva, Juan F. Spedaletti. Gamma convergence and asymptotic behavior for eigenvalues of nonlocal problems. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2125-2140. doi: 10.3934/dcds.2020355 |
[16] |
Fei-Ying Yang, Wan-Tong Li, Jian-Wen Sun. Principal eigenvalues for some nonlocal eigenvalue problems and applications. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 4027-4049. doi: 10.3934/dcds.2016.36.4027 |
[17] |
Jiří Benedikt. Continuous dependence of eigenvalues of $p$-biharmonic problems on $p$. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1469-1486. doi: 10.3934/cpaa.2013.12.1469 |
[18] |
Tomas Godoy, Jean-Pierre Gossez, Sofia Paczka. On the principal eigenvalues of some elliptic problems with large drift. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 225-237. doi: 10.3934/dcds.2013.33.225 |
[19] |
Michele Di Cristo. Stability estimates in the inverse transmission scattering problem. Inverse Problems and Imaging, 2009, 3 (4) : 551-565. doi: 10.3934/ipi.2009.3.551 |
[20] |
Kunquan Lan. Eigenvalues of second order differential equations with singularities. Conference Publications, 2001, 2001 (Special) : 241-247. doi: 10.3934/proc.2001.2001.241 |
2020 Impact Factor: 1.639
Tools
Metrics
Other articles
by authors
[Back to Top]