Spectral analysis of metamaterials in curved manifolds
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917867959615488 |
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| author | Faikl, Tomáš |
| author_facet | Faikl, Tomáš |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_10108 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectral analysis of metamaterials in curved manifolds Faikl, Tomáš Mathematical Physics 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. |
| title | Spectral analysis of metamaterials in curved manifolds |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2412.10108 |