American Institute of Mathematical Sciences

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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

Ha Pham 1, and  Plamen Stefanov 2,

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

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, Springer-Verlag, Berlin, 2006.  Google Scholar

[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.  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, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 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), 237-255. 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), 1472-1483. 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), 97-125. 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), 13-28. 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), 2965-2986. doi: 10.1137/100793748.  Google Scholar

[10]

________, The interior transmission problem and bounds on transmission eigenvalues, Math. Res. Lett., 18 (2011), 279-293. Google Scholar

[11]

________, Transmission eigenvalues for elliptic operators, SIAM J. Math. Anal., 43 (2011), 2630-2639. Google Scholar

[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.  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, Oxford University Press, Oxford, 2008.  Google Scholar

[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.  Google Scholar

[15]

________, Ellipticity in the interior transmission problem in anisotropic media, SIAM J. Math. Anal., 44 (2012), 1165-1174. Google Scholar

[16]

________, Remarks on interior transmission eigenvalues, Weyl formula and branching billiards, J. Phys. A, 45 (2012), 125202, 10 pp. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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), 351-382. 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, Publishers], New York-London, 1974. Computer Science and Applied Mathematics.  Google Scholar

[22]

L. Päivärinta and J. Sylvester, Transmission eigenvalues, SIAM J. Math. Anal., 40 (2008), 738-753. doi: 10.1137/070697525.  Google Scholar

[23]

V. Petkov and G. Vodev, Asymptotics of the number of the interior transmission eigenvalues, arXiv:1403.3949v2, (2014). Google Scholar

[24]

L. Robbiano, Counting function for interior transmission eigenvalues, arXiv:1310.6273, (2013). Google Scholar

[25]

________, Spectral analysis on interior transmission eigenvalues, arXiv:1302.4851, (2013). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

J. Sylvester, Discreteness of transmission eigenvalues via upper triangular compact operators, SIAM J. Math. Anal., 44 (2012), 341-354. doi: 10.1137/110836420.  Google Scholar

[30]

J. Sylvester, Transmission eigenvalues in one dimension, Inverse Problems, 29 (2013), 104009, 11 pp. 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), 283-302. 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), 277-296. doi: 10.1016/0022-1236(87)90069-3.  Google Scholar

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.  Google Scholar

[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.  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, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 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), 237-255. 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), 1472-1483. 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), 97-125. 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), 13-28. 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), 2965-2986. doi: 10.1137/100793748.  Google Scholar

[10]

________, The interior transmission problem and bounds on transmission eigenvalues, Math. Res. Lett., 18 (2011), 279-293. Google Scholar

[11]

________, Transmission eigenvalues for elliptic operators, SIAM J. Math. Anal., 43 (2011), 2630-2639. Google Scholar

[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.  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, Oxford University Press, Oxford, 2008.  Google Scholar

[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.  Google Scholar

[15]

________, Ellipticity in the interior transmission problem in anisotropic media, SIAM J. Math. Anal., 44 (2012), 1165-1174. Google Scholar

[16]

________, Remarks on interior transmission eigenvalues, Weyl formula and branching billiards, J. Phys. A, 45 (2012), 125202, 10 pp. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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), 351-382. 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, Publishers], New York-London, 1974. Computer Science and Applied Mathematics.  Google Scholar

[22]

L. Päivärinta and J. Sylvester, Transmission eigenvalues, SIAM J. Math. Anal., 40 (2008), 738-753. doi: 10.1137/070697525.  Google Scholar

[23]

V. Petkov and G. Vodev, Asymptotics of the number of the interior transmission eigenvalues, arXiv:1403.3949v2, (2014). Google Scholar

[24]

L. Robbiano, Counting function for interior transmission eigenvalues, arXiv:1310.6273, (2013). Google Scholar

[25]

________, Spectral analysis on interior transmission eigenvalues, arXiv:1302.4851, (2013). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

J. Sylvester, Discreteness of transmission eigenvalues via upper triangular compact operators, SIAM J. Math. Anal., 44 (2012), 341-354. doi: 10.1137/110836420.  Google Scholar

[30]

J. Sylvester, Transmission eigenvalues in one dimension, Inverse Problems, 29 (2013), 104009, 11 pp. 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), 283-302. 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), 277-296. doi: 10.1016/0022-1236(87)90069-3.  Google Scholar

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