American Institute of Mathematical Sciences

Advanced
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, (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

show all references

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

[1]

Vesselin Petkov, Georgi Vodev. Localization of the interior transmission eigenvalues for a ball. Inverse Problems & Imaging, 2017, 11 (2) : 355-372. doi: 10.3934/ipi.2017017
[2]

Luc Robbiano. Counting function for interior transmission eigenvalues. Mathematical Control & Related Fields, 2016, 6 (1) : 167-183. doi: 10.3934/mcrf.2016.6.167
[3]

David Colton, Lassi Päivärinta, John Sylvester. The interior transmission problem. Inverse Problems & Imaging, 2007, 1 (1) : 13-28. doi: 10.3934/ipi.2007.1.13
[4]

Kyoungsun Kim, Gen Nakamura, Mourad Sini. The Green function of the interior transmission problem and its applications. Inverse Problems & Imaging, 2012, 6 (3) : 487-521. doi: 10.3934/ipi.2012.6.487
[5]

Fioralba Cakoni, Houssem Haddar. A variational approach for the solution of the electromagnetic interior transmission problem for anisotropic media. Inverse Problems & Imaging, 2007, 1 (3) : 443-456. doi: 10.3934/ipi.2007.1.443
[6]

Andreas Kirsch. An integral equation approach and the interior transmission problem for Maxwell's equations. Inverse Problems & Imaging, 2007, 1 (1) : 159-179. doi: 10.3934/ipi.2007.1.159
[7]

Fang Zeng, Pablo Suarez, Jiguang Sun. A decomposition method for an interior inverse scattering problem. Inverse Problems & Imaging, 2013, 7 (1) : 291-303. doi: 10.3934/ipi.2013.7.291
[8]

Robert Carlson. Spectral theory for nonconservative transmission line networks. Networks & Heterogeneous Media, 2011, 6 (2) : 257-277. doi: 10.3934/nhm.2011.6.257
[9]

David Colton, Yuk-J. Leung. On a transmission eigenvalue problem for a spherically stratified coated dielectric. Inverse Problems & Imaging, 2016, 10 (2) : 369-378. doi: 10.3934/ipi.2016004
[10]

Jun Zhang, Xinyue Fan. An efficient spectral method for the Helmholtz transmission eigenvalues in polar geometries. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4799-4813. doi: 10.3934/dcdsb.2019031
[11]

Michele Di Cristo. Stability estimates in the inverse transmission scattering problem. Inverse Problems & Imaging, 2009, 3 (4) : 551-565. doi: 10.3934/ipi.2009.3.551
[12]

Fioralba Cakoni, Drossos Gintides. New results on transmission eigenvalues. Inverse Problems & Imaging, 2010, 4 (1) : 39-48. doi: 10.3934/ipi.2010.4.39
[13]

Andreas Kirsch. On the existence of transmission eigenvalues. Inverse Problems & Imaging, 2009, 3 (2) : 155-172. doi: 10.3934/ipi.2009.3.155
[14]

Fioralba Cakoni, Shari Moskow, Scott Rome. Asymptotic expansions of transmission eigenvalues for small perturbations of media with generally signed contrast. Inverse Problems & Imaging, 2018, 12 (4) : 971-992. doi: 10.3934/ipi.2018041
[15]

Brian Sleeman. The inverse acoustic obstacle scattering problem and its interior dual. Inverse Problems & Imaging, 2009, 3 (2) : 211-229. doi: 10.3934/ipi.2009.3.211
[16]

Fioralba Cakoni, Houssem Haddar, Isaac Harris. Homogenization of the transmission eigenvalue problem for periodic media and application to the inverse problem. Inverse Problems & Imaging, 2015, 9 (4) : 1025-1049. doi: 10.3934/ipi.2015.9.1025
[17]

Armin Lechleiter. The factorization method is independent of transmission eigenvalues. Inverse Problems & Imaging, 2009, 3 (1) : 123-138. doi: 10.3934/ipi.2009.3.123
[18]

Fioralba Cakoni, Anne Cossonnière, Houssem Haddar. Transmission eigenvalues for inhomogeneous media containing obstacles. Inverse Problems & Imaging, 2012, 6 (3) : 373-398. doi: 10.3934/ipi.2012.6.373
[19]

Yalin Zhang, Guoliang Shi. Continuous dependence of the transmission eigenvalues in one dimension. Inverse Problems & Imaging, 2015, 9 (1) : 273-287. doi: 10.3934/ipi.2015.9.273
[20]

Tiexiang Li, Tsung-Ming Huang, Wen-Wei Lin, Jenn-Nan Wang. On the transmission eigenvalue problem for the acoustic equation with a negative index of refraction and a practical numerical reconstruction method. Inverse Problems & Imaging, 2018, 12 (4) : 1033-1054. doi: 10.3934/ipi.2018043

2018 Impact Factor: 1.469

Tools

Metrics

  • PDF downloads (4)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences