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arXiv · 2412.10108

Spectral analysis of metamaterials in curved manifolds

Tomáš Faikl
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Abstract

Negative-index metamaterials possess a negative refractive index and thus present an interesting substance for designing uncommon optical effects such as invisibility cloaking. This paper deals with operators encountered in an operator-theoretic description of metamaterials. First, we introduce an indefinite Laplacian and consider it on a compact tubular neighbourhood in constantly curved compact two-dimensional Riemannian ambient manifolds, with Euclidean rectangle in $\mathbb{R}^2$ being present as a special case. As this operator is not semi-bounded, standard form-theoretic methods cannot be applied. We show that this operator is (essentially) self-adjoint via separation of variables and find its spectral characteristics. We also provide a new method for obtaining alternative definition of the self-adjoint operator in non-critical case via a generalized form representation theorem. The main motivation is existence of essential spectrum in bounded domains.

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Tomáš Faikl. 2024-12-13. Spectral analysis of metamaterials in curved manifolds. https://doi.org/10.1088/1751-8121%2Fad9dc5

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