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Detecting the localization of elastic inclusions and Lamé coefficients
Weyl asymptotics of the transmission eigenvalues for a constant index of refraction
1. | Department of Mathematics, Purdue University, West Lafayette, IN 47907, United States |
2. | Department of Mathematics, Purdue University, 150 N University Street, West Lafayette, IN 47907 |
References:
[1] |
F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory. An introduction, Interaction of Mechanics and Mathematics, Springer-Verlag, Berlin, 2006. |
[2] |
F. Cakoni, D. Colton and D. Gintides, The interior transmission eigenvalue problem, SIAM J. Math. Anal., 42 (2010), 2912-2921.
doi: 10.1137/100793542. |
[3] |
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, PA, 2011.
doi: 10.1137/1.9780898719406. |
[4] |
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. |
[5] |
D. Colton, A. Kirsch and L. Päivärinta, Far-field patterns for acoustic waves in an inhomogeneous medium, SIAM J. Math. Anal., 20 (1989), 1472-1483.
doi: 10.1137/0520096. |
[6] |
D. Colton and P. Monk, The inverse scattering problem for time-harmonic acoustic waves in an inhomogeneous medium, Quart. J. Mech. Appl. Math., 41 (1988), 97-125.
doi: 10.1093/qjmam/41.1.97. |
[7] |
D. Colton, L. Päivärinta and J. Sylvester, The interior transmission problem, Inverse Probl. Imaging, 1 (2007), 13-28.
doi: 10.3934/ipi.2007.1.13. |
[8] |
M. Dimassi and V. Petkov, Upper bound for the counting function of interior transmission eigenvalues, arXiv: math.SP, (2013). |
[9] |
M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta, Transmission eigenvalues for operators with constant coefficients, SIAM J. Math. Anal., 42 (2010), 2965-2986.
doi: 10.1137/100793748. |
[10] |
________, The interior transmission problem and bounds on transmission eigenvalues, Math. Res. Lett., 18 (2011), 279-293. |
[11] |
________, Transmission eigenvalues for elliptic operators, SIAM J. Math. Anal., 43 (2011), 2630-2639. |
[12] |
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. |
[13] |
A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, vol. 36 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2008. |
[14] |
E. Lakshtanov and B. Vainberg, Bounds on positive interior transmission eigenvalues, Inverse Problems, 28 (2012), 105005, 13 pp.
doi: 10.1088/0266-5611/28/10/105005. |
[15] |
________, Ellipticity in the interior transmission problem in anisotropic media, SIAM J. Math. Anal., 44 (2012), 1165-1174. |
[16] |
________, Remarks on interior transmission eigenvalues, Weyl formula and branching billiards, J. Phys. A, 45 (2012), 125202, 10 pp. |
[17] |
E. Lakshtanov and B. Vainberg, Applications of elliptic operator theory to the isotropic interior transmission eigenvalue problem, Inverse Problems, 29 (2013), 104003, 19 pp.
doi: 10.1088/0266-5611/29/10/104003. |
[18] |
E. Lakshtanov and B. Vainberg, Weyl type bound on positive interior transmission eigenvalues, Comm. Partial Differential Equations, 39 (2014), 1729-1740.
doi: 10.1080/03605302.2014.881853. |
[19] |
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. |
[20] |
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. |
[21] |
F. W. J. Olver, Asymptotics and Special Functions, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Computer Science and Applied Mathematics. |
[22] |
L. Päivärinta and J. Sylvester, Transmission eigenvalues, SIAM J. Math. Anal., 40 (2008), 738-753.
doi: 10.1137/070697525. |
[23] |
V. Petkov and G. Vodev, Asymptotics of the number of the interior transmission eigenvalues, arXiv:1403.3949v2, (2014). |
[24] |
L. Robbiano, Counting function for interior transmission eigenvalues, arXiv:1310.6273, (2013). |
[25] |
________, Spectral analysis on interior transmission eigenvalues, arXiv:1302.4851, (2013). |
[26] |
V. Serov and J. Sylvester, Transmission eigenvalues for degenerate and singular cases, Inverse Problems, 28 (2012), 065004, 8 pp.
doi: 10.1088/0266-5611/28/6/065004. |
[27] |
J. Sjöstrand and M. Zworski, Complex scaling and the distribution of scattering poles, J. Amer. Math. Soc., 4 (1991), 729-769.
doi: 10.1090/S0894-0347-1991-1115789-9. |
[28] |
P. Stefanov, Sharp upper bounds on the number of the scattering poles, J. Funct. Anal., 231 (2006), 111-142.
doi: 10.1016/j.jfa.2005.07.007. |
[29] |
J. Sylvester, Discreteness of transmission eigenvalues via upper triangular compact operators, SIAM J. Math. Anal., 44 (2012), 341-354.
doi: 10.1137/110836420. |
[30] |
J. Sylvester, Transmission eigenvalues in one dimension, Inverse Problems, 29 (2013), 104009, 11 pp.
doi: 10.1088/0266-5611/29/10/104009. |
[31] |
E. C. Titchmarsh, The Zeros of Certain Integral Functions, Proc. London Math. Soc., S2-25 (1926), 283-302.
doi: 10.1112/plms/s2-25.1.283. |
[32] |
M. Zworski, Distribution of poles for scattering on the real line, J. Funct. Anal., 73 (1987), 277-296.
doi: 10.1016/0022-1236(87)90069-3. |
show all references
References:
[1] |
F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory. An introduction, Interaction of Mechanics and Mathematics, Springer-Verlag, Berlin, 2006. |
[2] |
F. Cakoni, D. Colton and D. Gintides, The interior transmission eigenvalue problem, SIAM J. Math. Anal., 42 (2010), 2912-2921.
doi: 10.1137/100793542. |
[3] |
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, PA, 2011.
doi: 10.1137/1.9780898719406. |
[4] |
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. |
[5] |
D. Colton, A. Kirsch and L. Päivärinta, Far-field patterns for acoustic waves in an inhomogeneous medium, SIAM J. Math. Anal., 20 (1989), 1472-1483.
doi: 10.1137/0520096. |
[6] |
D. Colton and P. Monk, The inverse scattering problem for time-harmonic acoustic waves in an inhomogeneous medium, Quart. J. Mech. Appl. Math., 41 (1988), 97-125.
doi: 10.1093/qjmam/41.1.97. |
[7] |
D. Colton, L. Päivärinta and J. Sylvester, The interior transmission problem, Inverse Probl. Imaging, 1 (2007), 13-28.
doi: 10.3934/ipi.2007.1.13. |
[8] |
M. Dimassi and V. Petkov, Upper bound for the counting function of interior transmission eigenvalues, arXiv: math.SP, (2013). |
[9] |
M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta, Transmission eigenvalues for operators with constant coefficients, SIAM J. Math. Anal., 42 (2010), 2965-2986.
doi: 10.1137/100793748. |
[10] |
________, The interior transmission problem and bounds on transmission eigenvalues, Math. Res. Lett., 18 (2011), 279-293. |
[11] |
________, Transmission eigenvalues for elliptic operators, SIAM J. Math. Anal., 43 (2011), 2630-2639. |
[12] |
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. |
[13] |
A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, vol. 36 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2008. |
[14] |
E. Lakshtanov and B. Vainberg, Bounds on positive interior transmission eigenvalues, Inverse Problems, 28 (2012), 105005, 13 pp.
doi: 10.1088/0266-5611/28/10/105005. |
[15] |
________, Ellipticity in the interior transmission problem in anisotropic media, SIAM J. Math. Anal., 44 (2012), 1165-1174. |
[16] |
________, Remarks on interior transmission eigenvalues, Weyl formula and branching billiards, J. Phys. A, 45 (2012), 125202, 10 pp. |
[17] |
E. Lakshtanov and B. Vainberg, Applications of elliptic operator theory to the isotropic interior transmission eigenvalue problem, Inverse Problems, 29 (2013), 104003, 19 pp.
doi: 10.1088/0266-5611/29/10/104003. |
[18] |
E. Lakshtanov and B. Vainberg, Weyl type bound on positive interior transmission eigenvalues, Comm. Partial Differential Equations, 39 (2014), 1729-1740.
doi: 10.1080/03605302.2014.881853. |
[19] |
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. |
[20] |
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. |
[21] |
F. W. J. Olver, Asymptotics and Special Functions, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Computer Science and Applied Mathematics. |
[22] |
L. Päivärinta and J. Sylvester, Transmission eigenvalues, SIAM J. Math. Anal., 40 (2008), 738-753.
doi: 10.1137/070697525. |
[23] |
V. Petkov and G. Vodev, Asymptotics of the number of the interior transmission eigenvalues, arXiv:1403.3949v2, (2014). |
[24] |
L. Robbiano, Counting function for interior transmission eigenvalues, arXiv:1310.6273, (2013). |
[25] |
________, Spectral analysis on interior transmission eigenvalues, arXiv:1302.4851, (2013). |
[26] |
V. Serov and J. Sylvester, Transmission eigenvalues for degenerate and singular cases, Inverse Problems, 28 (2012), 065004, 8 pp.
doi: 10.1088/0266-5611/28/6/065004. |
[27] |
J. Sjöstrand and M. Zworski, Complex scaling and the distribution of scattering poles, J. Amer. Math. Soc., 4 (1991), 729-769.
doi: 10.1090/S0894-0347-1991-1115789-9. |
[28] |
P. Stefanov, Sharp upper bounds on the number of the scattering poles, J. Funct. Anal., 231 (2006), 111-142.
doi: 10.1016/j.jfa.2005.07.007. |
[29] |
J. Sylvester, Discreteness of transmission eigenvalues via upper triangular compact operators, SIAM J. Math. Anal., 44 (2012), 341-354.
doi: 10.1137/110836420. |
[30] |
J. Sylvester, Transmission eigenvalues in one dimension, Inverse Problems, 29 (2013), 104009, 11 pp.
doi: 10.1088/0266-5611/29/10/104009. |
[31] |
E. C. Titchmarsh, The Zeros of Certain Integral Functions, Proc. London Math. Soc., S2-25 (1926), 283-302.
doi: 10.1112/plms/s2-25.1.283. |
[32] |
M. Zworski, Distribution of poles for scattering on the real line, J. Funct. Anal., 73 (1987), 277-296.
doi: 10.1016/0022-1236(87)90069-3. |
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