Spectra of elliptic operators and applications

Abstract: In this thesis we consider several problems related to elliptic equations. Namely, we investigate Helmholtz and Maxwell's equations in Paper I and Sturm-Liouville spectral problem in Papers II,III. In Paper I we study time-harmonic electromagnetic and acoustic waveguide, modeled by an in�nite cylinder with a non-smooth cross section. We introduce an in- �nitesimal generator for the wave evolution along the cylinder, and prove estimates of the functional calculi of these �rst order non-self adjoint di�erential operators with non-smooth coe�cients. Applying our new functional calculus, we obtain a one-to-one correspondence between polynomially bounded time-harmonic waves and functions in appropriate spectral subspaces. In Paper II we �nd asymptotic formulas for the eigenvalues of the Sturm-Liouville operator on the �nite interval, with potential having a strong negative singularity at one endpoint. In Paper III we consider Sturm-Liouville operator with delta-interactions, two-sided estimates of the distribution function of the eigenvalues and a criterion for the discreteness of the spectrum in terms of the Otelbaev function are obtained. A criterion for the resolvent of the Sturm-Liouville operator to belong to the class Sp is established.