# American Institute of Mathematical Sciences

August  2018, 12(4): 971-992. doi: 10.3934/ipi.2018041

## Asymptotic expansions of transmission eigenvalues for small perturbations of media with generally signed contrast

 1 Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA 2 Department of Mathematics, Drexel University, Philadelphia, PA 19104, USA

Received  August 2017 Published  June 2018

In this paper we revisit the transmission eigenvalue problem for an inhomogeneous media of compact support perturbed by small penetrable homogeneous inclusions. Assuming that the inhomogeneous background media is known and smooth, we investigate how these small volume inclusions affect the transmission eigenvalues. Our perturbation analysis makes use of the formulation of the transmission eigenvalue problem introduced Kirsch in [8], which requires that the contrast of the inhomogeneity is of one-sign only near the boundary. Thus, our approach can handle small perturbations with positive, negative or zero (voids) contrasts. In addition to proving the convergence rate for the eigenvalues corresponding to the perturbed media as inclusions' volume goes to zero, we also provide the explicit first correction term in the asymptotic expansion for simple eigenvalues. The correction term involves computable information about the known inhomogeneity as well as the location, size and refractive index of small perturbations. Our asymptotic formula has the potential to be used to recover information about small inclusions from knowledge of the real transmission eigenvalues, which can be determined from scattering data.

Citation: 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
##### References:
 [1] F. Cakoni, D. Colton and H. Haddar, On the determination of Dirichlet or transmission eigenvalues from far field data, C. R. Math. Acad. Sci. Paris, 348 (2010), 379-383. doi: 10.1016/j.crma.2010.02.003. Google Scholar [2] F. Cakoni, D. Colton and H. Haddar, Inverse Scattering Theory and Transmission Eigenvalues, vol. 88 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2016. doi: 10.1137/1.9781611974461.ch1. Google Scholar [3] 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 [4] F. Cakoni, H. Harris and S. Moskow, The imaging of small perturbations in an anisotropic media, Computers and Mathematics with Applications, 74 (2017), 2769-2783. doi: 10.1016/j.camwa.2017.06.050. Google Scholar [5] F. Cakoni and S. Moskow, Asymptotic expansions for transmission eigenvalues for media with small inhomogeneities, Inverse Problems, 29 (2013), 104014, 18pp. doi: 10.1088/0266-5611/29/10/104014. Google Scholar [6] F. Cakoni, S. Moskow and S. Rome, The perturbation of transmission eigenvalues for inhomogeneous media in the presence of small penetrable inclusions, Inverse Probl. Imaging, 9 (2015), 725-748. doi: 10.3934/ipi.2015.9.725. Google Scholar [7] L. Hörmander, The Analysis of Linear Partial Differential Operators II: Differential Operators with Constant Coefficients, Springer-Verlag, Berlin, 2005. doi: 10.1007/b138375. Google Scholar [8] A. Kirsch, A note on Sylvester's proof of discreteness of interior transmission eigenvalues, C. R. Math. Acad. Sci. Paris, 354 (2016), 377-382. doi: 10.1016/j.crma.2016.01.015. Google Scholar [9] A. Kirsch and A. Lechleiter, The inside-outside duality for scattering problems by inhomogeneous media, Inverse Problems, 29 (2013), 104011, 21pp. doi: 10.1088/0266-5611/29/10/104011. Google Scholar [10] P. Monk, Finite Element Methods for Maxwell's Equations, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2003. doi: 10.1093/acprof:oso/9780198508885.001.0001. Google Scholar [11] S. Moskow, Nonlinear eigenvalue approximation for compact operators, J. Math. Phys., 56 (2015), 113512, 11pp. doi: 10.1063/1.4936304. Google Scholar [12] J. E. Osborn, Spectral approximation for compact operators, Math. Comput., 29 (1975), 712-725. doi: 10.1090/S0025-5718-1975-0383117-3. Google Scholar [13] V. Petkov and G. Vodev, Asymptotics of the number of the interior transmission eigenvalues, J. Spectr. Theory, 7 (2017), 1-31. doi: 10.4171/JST/154. Google Scholar [14] L. Robbiano, Spectral analysis of the interior transmission eigenvalue problem, Inverse Problems, 29 (2013), 104001, 28pp. doi: 10.1088/0266-5611/29/10/104001. Google Scholar [15] L. Robbiano, Counting function for interior transmission eigenvalues, Math. Control Relat. Fields, 6 (2016), 167-183. doi: 10.3934/mcrf.2016.6.167. Google Scholar [16] 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

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##### References:
 [1] F. Cakoni, D. Colton and H. Haddar, On the determination of Dirichlet or transmission eigenvalues from far field data, C. R. Math. Acad. Sci. Paris, 348 (2010), 379-383. doi: 10.1016/j.crma.2010.02.003. Google Scholar [2] F. Cakoni, D. Colton and H. Haddar, Inverse Scattering Theory and Transmission Eigenvalues, vol. 88 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2016. doi: 10.1137/1.9781611974461.ch1. Google Scholar [3] 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 [4] F. Cakoni, H. Harris and S. Moskow, The imaging of small perturbations in an anisotropic media, Computers and Mathematics with Applications, 74 (2017), 2769-2783. doi: 10.1016/j.camwa.2017.06.050. Google Scholar [5] F. Cakoni and S. Moskow, Asymptotic expansions for transmission eigenvalues for media with small inhomogeneities, Inverse Problems, 29 (2013), 104014, 18pp. doi: 10.1088/0266-5611/29/10/104014. Google Scholar [6] F. Cakoni, S. Moskow and S. Rome, The perturbation of transmission eigenvalues for inhomogeneous media in the presence of small penetrable inclusions, Inverse Probl. Imaging, 9 (2015), 725-748. doi: 10.3934/ipi.2015.9.725. Google Scholar [7] L. Hörmander, The Analysis of Linear Partial Differential Operators II: Differential Operators with Constant Coefficients, Springer-Verlag, Berlin, 2005. doi: 10.1007/b138375. Google Scholar [8] A. Kirsch, A note on Sylvester's proof of discreteness of interior transmission eigenvalues, C. R. Math. Acad. Sci. Paris, 354 (2016), 377-382. doi: 10.1016/j.crma.2016.01.015. Google Scholar [9] A. Kirsch and A. Lechleiter, The inside-outside duality for scattering problems by inhomogeneous media, Inverse Problems, 29 (2013), 104011, 21pp. doi: 10.1088/0266-5611/29/10/104011. Google Scholar [10] P. Monk, Finite Element Methods for Maxwell's Equations, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2003. doi: 10.1093/acprof:oso/9780198508885.001.0001. Google Scholar [11] S. Moskow, Nonlinear eigenvalue approximation for compact operators, J. Math. Phys., 56 (2015), 113512, 11pp. doi: 10.1063/1.4936304. Google Scholar [12] J. E. Osborn, Spectral approximation for compact operators, Math. Comput., 29 (1975), 712-725. doi: 10.1090/S0025-5718-1975-0383117-3. Google Scholar [13] V. Petkov and G. Vodev, Asymptotics of the number of the interior transmission eigenvalues, J. Spectr. Theory, 7 (2017), 1-31. doi: 10.4171/JST/154. Google Scholar [14] L. Robbiano, Spectral analysis of the interior transmission eigenvalue problem, Inverse Problems, 29 (2013), 104001, 28pp. doi: 10.1088/0266-5611/29/10/104001. Google Scholar [15] L. Robbiano, Counting function for interior transmission eigenvalues, Math. Control Relat. Fields, 6 (2016), 167-183. doi: 10.3934/mcrf.2016.6.167. Google Scholar [16] 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
Comparison of perturbed eigenvalues and corrected eigenvalues. The red circles are the perturbed transmission eigenvalues (squared) and the blue stars the corrected approximations for various values of $\epsilon$. The $x$-axis is $\log_{10}{\epsilon}$
Log/log plot of the error $(\tau_\epsilon-(\tau_0+\epsilon \tau^{(1)} ))/\epsilon$
Parameters for Numerical Example
 Domain $D$ $[-1, 1]$ Background Transmission Eigenvalue $k=\sqrt{-\tau}$ 7.12761 Background Coefficient $q_0$ 6.29 Perturbed Coefficient $q_1$ 24 Parameter $\lambda$ 50.72217
 Domain $D$ $[-1, 1]$ Background Transmission Eigenvalue $k=\sqrt{-\tau}$ 7.12761 Background Coefficient $q_0$ 6.29 Perturbed Coefficient $q_1$ 24 Parameter $\lambda$ 50.72217
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