Spectral analysis of metamaterials in curved manifolds

TL;DR

This paper analyzes the spectral properties of an indefinite Laplacian operator on curved manifolds, relevant for understanding negative-index metamaterials, and introduces new methods for defining self-adjoint operators in this context.

Contribution

It introduces a spectral analysis of indefinite Laplacians on curved manifolds and proposes a novel approach for defining self-adjoint operators without standard semi-boundedness assumptions.

Findings

Established self-adjointness of the indefinite Laplacian via separation of variables

Characterized the spectral properties of the operator on curved manifolds

Developed a new method for defining self-adjoint operators in non-critical cases

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.