Ground states and spectral properties in quantum field theories

Abstract

In this thesis we consider non-relativistic quantum electrodynamics in dipole approximation and study low-energy phenomenons of quantum mechanical systems. We investigate the analytic dependence of the lowest-energy eigenvalue and eigenvector on spectral parameters of the system. In particular we study situations where the ground-state eigenvalue is assumed to be degenerate. In the first situation the eigenspace of a degenerate ground-state eigenvalue is assumed to split up in a specific way in second order formal perturbation theory. We show, using a mild infrared assumption, that the emerging unique ground state and the corresponding ground-state eigenvalue are analytic functions of the coupling constant in a cone with apex at the origin. Secondly we analyse the situation that the degeneracy is protected by a set of symmetries for the considered quantum mechanical system. We prove, in accordance with known results for the non-degenerate situation, that the ground-state eigenvalue and eigenvectors depend analytically on the coupling constant. In order to show these results we extend operator-theoretic renormalization to such degenerate situations. To complement the analyticity results we additionally show that an asymptotic expansion of the ground state and the ground-state eigenvalue exists up to arbitrary order. The infrared assumption needed for the asymptotic expansion is weaker than the usual assumptions required for other methods such as operator theoretic renormalization to be applicable.