Perturbation Theory of Embedded Eigenvalues

By Matthias Engelmann
PhD Dissertations
December 2014
Abstract:
In this thesis problems connected with perturbation theory of embedded eigenvalues are studied. The first part deals with the case of the translation invariant massive Nelson model and establishes that embedded eigenvalues of the fiber Hamiltonians have to depend at least twice differentiable on the fiber parameter. This is done by deriving an expansion to order $2 + \alpha$ w.r.t. the fiber parameter. The main technical obstacles are that a priori the fiber Hamiltonians do not have sufficient regularity w.r.t. the conjugate operator and that commutators of the conjugate operator with the fiber Hamiltonians cannot be bounded by the Hamiltonians again. In the second part an abstract analytic perturbation theory for embedded eigenvalues is established. However a technical requirement on the Hamiltonians is that all iterated commutators should be bounded by the Hamiltonian thus excluding the Nelson model. The strategy of the paper is to use spectral deformation techniques were the unitary group is generated by the conjugate operator. A Mourre estimate allows to prove that the essential spectrum of the transformed Hamiltonians recedes thus permitting the use of Kato theory