A Class of Representations of the ∗-Algebra of the Canonical Commutation Relations over a Hilbert Space and Instability of Embedded Eigenvalues in Quantum Field Models

Abstract

In models of a quantum harmonic oscillator coupled to a quantum field with a quadratic interaction, embedded eigenvalues of the unperturbed system may be unstable under the perturbation given by the interaction of the oscillator with the quantum field. A general mathematical structure underlying this phenomenon is clarified in terms of a class of Fock space representations of the ∗-algebra of the canonical commutation relations over a Hilbert space. It is also shown that each of the representations is given as a composition of a proper Bogolyubov (canonical) transformation and a partial isometry on the Fock space of the representation.