Embedded eigenvalues for asymptotically periodic ODE Systems

Sammanfattning: In this thesis we investigate the persistance of embedded eigenvalues under perturbations of a certain self-adjoint Schrödinger-type differential operator in L^2(\mathbb{R},\mathbb{R}^n), with an asymptotically periodic potential. The studied perturbations are small and belong to a certain Banach space with a specified decay rate, in particular, a weighted space of continuous matrix valued functions. The set of perturbations for which the embedded eigenvalue persists is shown to form a smooth manifold with a specified co-dimension. This is done using tools from Floquet theory, basic Banach space calculus, exponential dichotomies and their roughness properties, and Lyapunov-Schmidt reduction. In the end, as a way of showing that the investigated setting exists, a concrete example is presented. The example itself relates to a problem from quantum mechanics and represents a system of electrons in an infinite one-dimensional crystal.