Weyl asymptotics of the transmission eigenvalues for a constant index of refraction

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August 2014, 8(3): 795-810. doi: 10.3934/ipi.2014.8.795

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

Received October 2013 Revised April 2014 Published August 2014

Abstract
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We prove Weyl-type asymptotic formulas for the real and the complex internal transmission eigenvalues when the domain is a ball and the index of refraction is constant.

Keywords: Interior transmission problem, spectral theory, interior transmission eigenvalue, scattering., asymptotic distribution of eigenvalues.

Mathematics Subject Classification: Primary: 35P20, 47A75; Secondary: 35J40, 35P2.

Citation: Ha Pham, Plamen Stefanov. Weyl asymptotics of the transmission eigenvalues for a constant index of refraction. Inverse Problems & Imaging, 2014, 8 (3) : 795-810. doi: 10.3934/ipi.2014.8.795

References:

[1]	F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory. An introduction,, Interaction of Mechanics and Mathematics, (2006). Google Scholar
[2]	F. Cakoni, D. Colton and D. Gintides, The interior transmission eigenvalue problem,, SIAM J. Math. Anal., 42 (2010), 2912. doi: 10.1137/100793542. Google Scholar
[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, (2011). doi: 10.1137/1.9780898719406. Google Scholar
[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. doi: 10.1137/090769338. Google Scholar
[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. doi: 10.1137/0520096. Google Scholar
[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. doi: 10.1093/qjmam/41.1.97. Google Scholar
[7]	D. Colton, L. Päivärinta and J. Sylvester, The interior transmission problem,, Inverse Probl. Imaging, 1 (2007), 13. doi: 10.3934/ipi.2007.1.13. Google Scholar
[8]	M. Dimassi and V. Petkov, Upper bound for the counting function of interior transmission eigenvalues,, arXiv: math.SP, (2013). Google Scholar
[9]	M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta, Transmission eigenvalues for operators with constant coefficients,, SIAM J. Math. Anal., 42 (2010), 2965. doi: 10.1137/100793748. Google Scholar
[10]	________, The interior transmission problem and bounds on transmission eigenvalues,, Math. Res. Lett., 18 (2011), 279. Google Scholar
[11]	________, Transmission eigenvalues for elliptic operators,, SIAM J. Math. Anal., 43 (2011), 2630. Google Scholar
[12]	A. Kirsch, The denseness of the far field patterns for the transmission problem,, IMA J. Appl. Math., 37 (1986), 213. doi: 10.1093/imamat/37.3.213. Google Scholar
[13]	A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems,, vol. 36 of Oxford Lecture Series in Mathematics and its Applications, (2008). Google Scholar
[14]	E. Lakshtanov and B. Vainberg, Bounds on positive interior transmission eigenvalues,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/10/105005. Google Scholar
[15]	________, Ellipticity in the interior transmission problem in anisotropic media,, SIAM J. Math. Anal., 44 (2012), 1165. Google Scholar
[16]	________, Remarks on interior transmission eigenvalues, Weyl formula and branching billiards,, J. Phys. A, 45 (2012). Google Scholar
[17]	E. Lakshtanov and B. Vainberg, Applications of elliptic operator theory to the isotropic interior transmission eigenvalue problem,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/10/104003. Google Scholar
[18]	E. Lakshtanov and B. Vainberg, Weyl type bound on positive interior transmission eigenvalues,, Comm. Partial Differential Equations, 39 (2014), 1729. doi: 10.1080/03605302.2014.881853. Google Scholar
[19]	Y.-J. Leung and D. Colton, Complex transmission eigenvalues for spherically stratified media,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/7/075005. Google Scholar
[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. doi: 10.1006/jdeq.1994.1017. Google Scholar*
[21]	F. W. J. Olver, Asymptotics and Special Functions,, Academic Press [A subsidiary of Harcourt Brace Jovanovich, (1974). Google Scholar
[22]	L. Päivärinta and J. Sylvester, Transmission eigenvalues,, SIAM J. Math. Anal., 40 (2008), 738. doi: 10.1137/070697525. Google Scholar
[23]	V. Petkov and G. Vodev, Asymptotics of the number of the interior transmission eigenvalues,, , (2014). Google Scholar
[24]	L. Robbiano, Counting function for interior transmission eigenvalues,, , (2013). Google Scholar
[25]	________, Spectral analysis on interior transmission eigenvalues,, , (2013). Google Scholar
[26]	V. Serov and J. Sylvester, Transmission eigenvalues for degenerate and singular cases,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/6/065004. Google Scholar
[27]	J. Sjöstrand and M. Zworski, Complex scaling and the distribution of scattering poles,, J. Amer. Math. Soc., 4 (1991), 729. doi: 10.1090/S0894-0347-1991-1115789-9. Google Scholar
[28]	P. Stefanov, Sharp upper bounds on the number of the scattering poles,, J. Funct. Anal., 231* (2006), 111. doi: 10.1016/j.jfa.2005.07.007. Google Scholar*
[29]	J. Sylvester, Discreteness of transmission eigenvalues via upper triangular compact operators,, SIAM J. Math. Anal., 44 (2012), 341. doi: 10.1137/110836420. Google Scholar
[30]	J. Sylvester, Transmission eigenvalues in one dimension,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/10/104009. Google Scholar
[31]	E. C. Titchmarsh, The Zeros of Certain Integral Functions,, Proc. London Math. Soc., S2-25* (1926), 2. doi: 10.1112/plms/s2-25.1.283. Google Scholar*
[32]	M. Zworski, Distribution of poles for scattering on the real line,, J. Funct. Anal., 73 (1987), 277. doi: 10.1016/0022-1236(87)90069-3. Google Scholar

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References:

[1]	F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory. An introduction,, Interaction of Mechanics and Mathematics, (2006). Google Scholar
[2]	F. Cakoni, D. Colton and D. Gintides, The interior transmission eigenvalue problem,, SIAM J. Math. Anal., 42 (2010), 2912. doi: 10.1137/100793542. Google Scholar
[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, (2011). doi: 10.1137/1.9780898719406. Google Scholar
[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. doi: 10.1137/090769338. Google Scholar
[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. doi: 10.1137/0520096. Google Scholar
[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. doi: 10.1093/qjmam/41.1.97. Google Scholar
[7]	D. Colton, L. Päivärinta and J. Sylvester, The interior transmission problem,, Inverse Probl. Imaging, 1 (2007), 13. doi: 10.3934/ipi.2007.1.13. Google Scholar
[8]	M. Dimassi and V. Petkov, Upper bound for the counting function of interior transmission eigenvalues,, arXiv: math.SP, (2013). Google Scholar
[9]	M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta, Transmission eigenvalues for operators with constant coefficients,, SIAM J. Math. Anal., 42 (2010), 2965. doi: 10.1137/100793748. Google Scholar
[10]	________, The interior transmission problem and bounds on transmission eigenvalues,, Math. Res. Lett., 18 (2011), 279. Google Scholar
[11]	________, Transmission eigenvalues for elliptic operators,, SIAM J. Math. Anal., 43 (2011), 2630. Google Scholar
[12]	A. Kirsch, The denseness of the far field patterns for the transmission problem,, IMA J. Appl. Math., 37 (1986), 213. doi: 10.1093/imamat/37.3.213. Google Scholar
[13]	A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems,, vol. 36 of Oxford Lecture Series in Mathematics and its Applications, (2008). Google Scholar
[14]	E. Lakshtanov and B. Vainberg, Bounds on positive interior transmission eigenvalues,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/10/105005. Google Scholar
[15]	________, Ellipticity in the interior transmission problem in anisotropic media,, SIAM J. Math. Anal., 44 (2012), 1165. Google Scholar
[16]	________, Remarks on interior transmission eigenvalues, Weyl formula and branching billiards,, J. Phys. A, 45 (2012). Google Scholar
[17]	E. Lakshtanov and B. Vainberg, Applications of elliptic operator theory to the isotropic interior transmission eigenvalue problem,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/10/104003. Google Scholar
[18]	E. Lakshtanov and B. Vainberg, Weyl type bound on positive interior transmission eigenvalues,, Comm. Partial Differential Equations, 39 (2014), 1729. doi: 10.1080/03605302.2014.881853. Google Scholar
[19]	Y.-J. Leung and D. Colton, Complex transmission eigenvalues for spherically stratified media,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/7/075005. Google Scholar
[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. doi: 10.1006/jdeq.1994.1017. Google Scholar*
[21]	F. W. J. Olver, Asymptotics and Special Functions,, Academic Press [A subsidiary of Harcourt Brace Jovanovich, (1974). Google Scholar
[22]	L. Päivärinta and J. Sylvester, Transmission eigenvalues,, SIAM J. Math. Anal., 40 (2008), 738. doi: 10.1137/070697525. Google Scholar
[23]	V. Petkov and G. Vodev, Asymptotics of the number of the interior transmission eigenvalues,, , (2014). Google Scholar
[24]	L. Robbiano, Counting function for interior transmission eigenvalues,, , (2013). Google Scholar
[25]	________, Spectral analysis on interior transmission eigenvalues,, , (2013). Google Scholar
[26]	V. Serov and J. Sylvester, Transmission eigenvalues for degenerate and singular cases,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/6/065004. Google Scholar
[27]	J. Sjöstrand and M. Zworski, Complex scaling and the distribution of scattering poles,, J. Amer. Math. Soc., 4 (1991), 729. doi: 10.1090/S0894-0347-1991-1115789-9. Google Scholar
[28]	P. Stefanov, Sharp upper bounds on the number of the scattering poles,, J. Funct. Anal., 231* (2006), 111. doi: 10.1016/j.jfa.2005.07.007. Google Scholar*
[29]	J. Sylvester, Discreteness of transmission eigenvalues via upper triangular compact operators,, SIAM J. Math. Anal., 44 (2012), 341. doi: 10.1137/110836420. Google Scholar
[30]	J. Sylvester, Transmission eigenvalues in one dimension,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/10/104009. Google Scholar
[31]	E. C. Titchmarsh, The Zeros of Certain Integral Functions,, Proc. London Math. Soc., S2-25* (1926), 2. doi: 10.1112/plms/s2-25.1.283. Google Scholar*
[32]	M. Zworski, Distribution of poles for scattering on the real line,, J. Funct. Anal., 73 (1987), 277. doi: 10.1016/0022-1236(87)90069-3. Google Scholar

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