In this dissertation, we consider some spectral problems of optical waveguide and quantum graph theories. We study spectral problems that arise when considerating optical waveguides in photonic band-gap (PBG) materials. Specifically, we address the issue of the existence of modes guided by linear defects in photonic crystals. Such modes can be created for frequencies in the spectral gaps of the bulk material and thus are evanescent in the bulk (i.e., confined to the guide). In the quantum graph part we prove the validity of the limiting absorption principle for finite graphs with infinite leads attached. In particular, this leads to the absence of a singular continuous spectrum. Another problem in quantum graph theory that we consider involves opening gaps in the spectrum of a quantum graph by replacing each vertex of the original graph with a finite graph. We show that such "decorations" can be used to create spectral gaps.

In this dissertation, we consider some spectral problems of optical waveguide and quantum graph theories. We study spectral problems that arise when considerating optical waveguides in photonic band-gap (PBG) materials. Specifically, we address the issue of the existence of modes guided by linear defects in photonic crystals. Such modes can be created for frequencies in the spectral gaps of the bulk material and thus are evanescent in the bulk (i.e., confined to the guide). In the quantum graph part we prove the validity of the limiting absorption principle for finite graphs with infinite leads attached. In particular, this leads to the absence of a singular continuous spectrum. Another problem in quantum graph theory that we consider involves opening gaps in the spectrum of a quantum graph by replacing each vertex of the original graph with a finite graph. We show that such "decorations" can be used to create spectral gaps.