We consider two different models of a hydrogenic atom in a quantized electromagnetic field that treat the electron relativistically. The first one is a no-pair model in the free picture, the second one is given by the semi-relativistic Pauli-Fierz Hamiltonian. We prove that the no-pair operator is semi-bounded below and that its spectral subspaces corresponding to energies below the ionization threshold are exponentially localized. Both results hold true, for arbitrary values of the fine-structure constant, e 2 , and the ultra-violet cut-off, Λ, and for all nuclear charges less than the critical charge without radiation field, Z c = e −2 2/(2/π + π/2). We obtain similar results for the semi-relativistic Pauli-Fierz operator, again for all values of e 2 and Λ and for nuclear charges less than e −2 2/π.

We consider a hydrogen-like atom in a quantized electromagnetic field which is modeled by means of the semi-relativistic Pauli-Fierz operator and prove that the infimum of the spectrum of the latter operator is an eigenvalue. In particular, we verify that the bottom of its spectrum is strictly less than its ionization threshold. These results hold true for arbitrary values of the fine-structure constant and the ultra-violet cut-off as long as the Coulomb coupling constant (i.e. the product of the fine-structure constant and the nuclear charge) is less than 2/π.

We consider a hydrogen-like atom in a quantized electromagnetic field which is modeled by means of a no-pair operator acting in the positive spectral subspace of the free Dirac operator minimally coupled to the quantized vector potential. We prove that the infimum of the spectrum of the no-pair operator is an evenly degenerate eigenvalue. In particular, we show that the bottom of its spectrum is strictly less than its ionization threshold. These results hold true, for arbitrary values of the fine-structure constant and the ultra-violet cut-off and for all Coulomb coupling constants less than the critical one of the Brown-Ravenhall model, 2/(2/π + π/2). For Coulomb coupling constants larger than the critical one, we show that the quadratic form of the no-pair operator is unbounded below. Along the way we discuss the domains and operator cores of the semi-relativistic Pauli-Fierz and no-pair operators, for Coulomb coupling constants less than or equal to the critical ones. K m := L 2 (A m × 2 , dk) , dk := λ∈ 2 Am d 3 k , A m := {|k| m} .

We discuss Hilbert space-valued stochastic differential equations associated with the heat semi-groups of the standard model of non-relativistic quantum electrodynamics and of corresponding fiber Hamiltonians for translation invariant systems. In particular, we prove the existence of a stochastic flow satisfying the strong Markov property and the Feller property. To this end we employ an explicit solution ansatz. In the matrix-valued case, i.e., if the electron spin is taken into account, it is given by a series of operator-valued time-ordered integrals, whose integrands are factorized into annihilation, preservation, creation, and scalar parts. The Feynman-Kac formula implied by these results is new in the matrix-valued case. Furthermore, we discuss stochastic differential equations and Feynman-Kac representations for an operator-valued integral kernel of the semi-group. As a byproduct we obtain analogous results for Nelson's model.