A vanishing theorem for operators in Fock space

We consider the bosonic Fock space over the Hilbert space of transversal vector fields in three dimensions. This space carries a canonical representation of the group of rotations. For a certain class of operators in Fock space we show that rotation invariance implies the absence of terms which either create or annihilate only a single particle. We outline an application of this result in an operator theoretic renormalization analysis of Hamilton operators, which occur in non-relativistic qed.Comment: 14 page

Analytic Perturbation Theory and Renormalization Analysis of Matter Coupled to Quantized Radiation

For a large class of quantum mechanical models of matter and radiation we develop an analytic perturbation theory for non-degenerate ground states. This theory is applicable, for example, to models of matter with static nuclei and non-relativistic electrons that are coupled to the UV-cutoff quantized radiation field in the dipole approximation. If the lowest point of the energy spectrum is a non-degenerate eigenvalue of the Hamiltonian, we show that this eigenvalue is an analytic function of the nuclear coordinates and of $\alpha^{3/2}$, $\alpha$ being the fine structure constant. A suitably chosen ground state vector depends analytically on $\alpha^{3/2}$ and it is twice continuously differentiable with respect to the nuclear coordinates.Comment: 47 page

Smoothness and analyticity of perturbation expansions in QED

We consider the ground state of an atom in the framework of non-relativistic qed. We assume that the ultraviolet cutoff is of the order of the Rydberg energy and that the atomic Hamiltonian has a non-degenerate ground state. We show that the ground state energy and the ground state are k-times continuously differentiable functions of the fine structure constant and respectively the square root of the fine structure constant on some nonempty interval [0,c_k).Comment: 53 page

Convergent expansions in non-relativistic QED: Analyticity of the ground state

We consider the ground state of an atom in the framework of non-relativistic qed. We show that the ground state as well as the ground state energy are analytic functions of the coupling constant which couples to the vector potential, under the assumption that the atomic Hamiltonian has a non-degenerate ground state. Moreover, we show that the corresponding expansion coefficients are precisely the coefficients of the associated Raleigh-Schroedinger series. As a corollary we obtain that in a scaling limit where the ultraviolet cutoff is of the order of the Rydberg energy the ground state and the ground state energy have convergent power series expansions in the fine structure constant $\alpha$, with $\alpha$ dependent coefficients which are finite for $\alpha \geq 0$.Comment: 37 page

Ground state properties in non-relativistic QED

We discuss recent results concerning the ground state of non-relativistic quantum electrodynamics as a function of a magnetic coupling constant or the fine structure constant, obtained by the authors in [12,13,14].Comment: 6 Pages, contribution to the Proceedings of the Conference QMath 11 held in Hradec Kralove (Czechia) in September 201