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April  2017, 11(2): 355-372. doi: 10.3934/ipi.2017017

Localization of the interior transmission eigenvalues for a ball

Vesselin Petkov 1, and  Georgi Vodev 2,

1. 

Université de Bordeaux, Institut de Mathématiques de Bordeaux, 351, Cours de la Libération, 33405 Talence, France
2. 

Université de Nantes, Laboratoire de Mathématiques Jean Leray, 2, rue de la Houssiniére, BP 92208,44322 Nantes Cedex, France

Received  March 2016 Revised  January 2017 Published  March 2017

We study the localization of the interior transmission eigenvalues (ITEs) in the case when the domain is the unit ball $\{x ∈ \mathbb{R}^d:\: |x| ≤ 1\}, \: d≥ 2,$ and the coefficients $c_j(x), \: j =1,2, $ and the indices of refraction $n_j(x), \: j =1,2,$ are constants near the boundary $|x| = 1$. We prove that in this case the eigenvalue-free region obtained in [17] for strictly concave domains can be significantly improved. In particular, if $c_j(x), n_j(x), j = 1,2$ are constants for $|x| ≤ 1$, we show that all (ITEs) lie in a strip $|\operatorname{Im} λ| ≤ C$.

Keywords: Interior transmission eigenvalues, eigenvalue-free regions, interior Dirichlet-to-Neumann map, Bessel functions.
Mathematics Subject Classification: Primary: 35P15; Secondary: 35P20, 35P25.
Citation: 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
References:
[1]

F. Cakoni and H. Haddar, Transmission eigenvalues in inverse scattering theory, in Inverse Problems and Applications, Inside Out Ⅱ, G. Uhlmann, editor, MSRI Publications, Cambridge University Press, Cambridge, 60 (2013), 529-580.  Google Scholar

[2]

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

[3]

D. Colton and Y. -J. Leung, Complex eigenvalues and the inverse spectral problem for transmission eigenvalues Inverse Problems, 29 (2013), 104008, 6pp. doi: 10.1088/0266-5611/29/10/104008.  Google Scholar

[4]

D. Colton, Y. -J. Leung and S. Meng, Distribution of complex transmission eigenvalues for spherically stratified media Inverse Problems, 31 (2015), 035006, 19pp. doi: 10.1088/0266-5611/31/3/035006.  Google Scholar

[5]

M. Faierman, The interior transmission problem: Spectral theory, SIAM J. Math. Anal., 46 (2014), 803-819.  doi: 10.1137/130922215.  Google Scholar

[6]

M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta, The interior transmission problem and bounds of transmission eigenvalues, Math. Res. Lett., 18 (2011), 279-293.  doi: 10.4310/MRL.2011.v18.n2.a7.  Google Scholar

[7]

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

[8]

E. Lakshtanov and B. Vainberg, Application of elliptic theory to the isotropic interior transmission eigenvalue problem Inverse Problems, 29 (2013), 104003, 19pp. doi: 10.1088/0266-5611/29/10/104003.  Google Scholar

[9]

Y. -J. Leung and D. Colton, Complex transmission eigenvalues for spherically stratified media Inverse Problems, 28 (2012), 075005, 9pp. doi: 10.1088/0266-5611/28/7/075005.  Google Scholar

[10]

F. Olver, Asymptotics and Special Functions, Academic Press, New York, London, 1974.  Google Scholar

[11]

H. Pham and P. Stefanov, Weyl asymptotics of the transmission eigenvalues for a constant index of refraction, Inverse Problems and Imaging, 8 (2014), 795-810.  doi: 10.3934/ipi.2014.8.795.  Google Scholar

[12]

L. Robbiano, Spectral analysis of interior transmission eigenvalues Inverse Problems, 29 (2013), 104001, 28pp. doi: 10.1088/0266-5611/29/10/104001.  Google Scholar

[13]

L. Robbiano, Counting function for interior transmission eigenvalues, Mathematical Control and Related Fields, 6 (2016), 167-183.  doi: 10.3934/mcrf.2016.6.167.  Google Scholar

[14]

J. Sylvester, Transmission eigenvalues in one dimension Inverse Problems, 29 (2013), 104009, 11pp. doi: 10.1088/0266-5611/29/10/104009.  Google Scholar

[15]

V. Petkov and G. Vodev, Asymptotics of the number of the interior transmission eigenvalues, J. Spectral Theory, to appear. Google Scholar

[16]

G. Vodev, Transmission eigenvalue-free regions, Comm. Math. Phys., 336 (2015), 1141-1166.  doi: 10.1007/s00220-015-2311-2.  Google Scholar

[17]

G. Vodev, Transmission eigenvalues for strictly concave domains, Math. Ann., 366 (2016), 301-336.  doi: 10.1007/s00208-015-1329-2.  Google Scholar

show all references

References:
[1]

F. Cakoni and H. Haddar, Transmission eigenvalues in inverse scattering theory, in Inverse Problems and Applications, Inside Out Ⅱ, G. Uhlmann, editor, MSRI Publications, Cambridge University Press, Cambridge, 60 (2013), 529-580.  Google Scholar

[2]

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

[3]

D. Colton and Y. -J. Leung, Complex eigenvalues and the inverse spectral problem for transmission eigenvalues Inverse Problems, 29 (2013), 104008, 6pp. doi: 10.1088/0266-5611/29/10/104008.  Google Scholar

[4]

D. Colton, Y. -J. Leung and S. Meng, Distribution of complex transmission eigenvalues for spherically stratified media Inverse Problems, 31 (2015), 035006, 19pp. doi: 10.1088/0266-5611/31/3/035006.  Google Scholar

[5]

M. Faierman, The interior transmission problem: Spectral theory, SIAM J. Math. Anal., 46 (2014), 803-819.  doi: 10.1137/130922215.  Google Scholar

[6]

M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta, The interior transmission problem and bounds of transmission eigenvalues, Math. Res. Lett., 18 (2011), 279-293.  doi: 10.4310/MRL.2011.v18.n2.a7.  Google Scholar

[7]

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

[8]

E. Lakshtanov and B. Vainberg, Application of elliptic theory to the isotropic interior transmission eigenvalue problem Inverse Problems, 29 (2013), 104003, 19pp. doi: 10.1088/0266-5611/29/10/104003.  Google Scholar

[9]

Y. -J. Leung and D. Colton, Complex transmission eigenvalues for spherically stratified media Inverse Problems, 28 (2012), 075005, 9pp. doi: 10.1088/0266-5611/28/7/075005.  Google Scholar

[10]

F. Olver, Asymptotics and Special Functions, Academic Press, New York, London, 1974.  Google Scholar

[11]

H. Pham and P. Stefanov, Weyl asymptotics of the transmission eigenvalues for a constant index of refraction, Inverse Problems and Imaging, 8 (2014), 795-810.  doi: 10.3934/ipi.2014.8.795.  Google Scholar

[12]

L. Robbiano, Spectral analysis of interior transmission eigenvalues Inverse Problems, 29 (2013), 104001, 28pp. doi: 10.1088/0266-5611/29/10/104001.  Google Scholar

[13]

L. Robbiano, Counting function for interior transmission eigenvalues, Mathematical Control and Related Fields, 6 (2016), 167-183.  doi: 10.3934/mcrf.2016.6.167.  Google Scholar

[14]

J. Sylvester, Transmission eigenvalues in one dimension Inverse Problems, 29 (2013), 104009, 11pp. doi: 10.1088/0266-5611/29/10/104009.  Google Scholar

[15]

V. Petkov and G. Vodev, Asymptotics of the number of the interior transmission eigenvalues, J. Spectral Theory, to appear. Google Scholar

[16]

G. Vodev, Transmission eigenvalue-free regions, Comm. Math. Phys., 336 (2015), 1141-1166.  doi: 10.1007/s00220-015-2311-2.  Google Scholar

[17]

G. Vodev, Transmission eigenvalues for strictly concave domains, Math. Ann., 366 (2016), 301-336.  doi: 10.1007/s00208-015-1329-2.  Google Scholar

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