Spectral analysis of metamaterials in curved manifolds

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.

In Journal of Physics A: Mathematical and Theoretical. IOP Publishing (2024)

References in corpus (1)

• Resolvent estimates for one-dimensional Schrödinger operators with complex potentials

Cited by in corpus (1)

• On two-dimensional Dirac operators with critical delta-shell interactions