Metamaterials: spectral-theoretic approach

This thesis is focused on metamaterial with simultaneously negative permittivity and permeability and their application in cloaking devices. Historical review of the inventing these substances both in practical and theoretical way is made here. We summarize the main properties and applications for the metamaterials and then focus mostly on the cloaking. Some treatments to this invisibility effect are mentioned but the main concern is here in concept of so called anomalous localised resonance. Freely inspired by it we confirm the proof that cloaking due to anomalous localised resonance does not occur for the three dimensional ball and extend this result for higher dimensions. Using operator theory we introduce an indefinite laplacian on rectangle and in the symmetric radial geometry and prove that both these operators are essentially self-adjoint. After that it is discovered that 0 lies in the essential spectra of both these operators which guarantee the existence of the inverse operator but not for all functions on the right side of Poisson equation.