Abstract
The modified transmission eigenvalue problem arises in inverse scattering theory for inhomogeneous media, by embedding the relevant medium into an other, artificially introduced inhomogeneous medium. In the present work, we examine the case where the artificial medium is characterised as a metamaterial, i.e. having a negative valued refractive index. Our aim is to construct an appropriate spectral Galerkin method to compute the modified transmission eigenvalues, with potential use to the inverse spectral problem as well. We show that the modified transmission eigenvalue problem corresponds to a compact and self-adjoint operator for which the eigenfuction system is not complete in the solution space. By introducing an auxiliary Dirichlet–Neumann eigenvalue problem, we construct an eigenfuction system which has the desired completeness property. We use this complete system to define the Galerkin scheme and by applying some results for compact and positive operator eigenvalue problems, we prove the convergence of our method. We present some numerical examples which validate the eigenvalues approximation. Finally, we pose the corresponding inverse spectral problem and show that the largest eigenvalue can determine an unknown constant refractive index.
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This research is carried out/funded in the context of the project “Direct and Inverse Spectral Problems in Scattering Theory” (MIS 5049186) under the call for proposals “Researchers’ support with an emphasis on young researchers- 2nd Cycle”. The project is co-financed by Greece and the European Union (European Social Fund- ESF) by the Operational Programme Human Resources Development, Education and Lifelong Learning 2014–2020.
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Gintides, D., Pallikarakis, N. & Stratouras, K. On the modified transmission eigenvalue problem with an artificial metamaterial background. Res Math Sci 8, 40 (2021). https://doi.org/10.1007/s40687-021-00278-z
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DOI: https://doi.org/10.1007/s40687-021-00278-z
Keywords
- Transmission eigenvalues
- Inhomogeneous medium
- Metamaterial refractive index
- Modified far field equations
- Galerkin approximation