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 ...

We consider the semi-flow defined by semi-linear parabolic equations in Lp-spaces and study the differentiable dependence on the initial data. Together with a spectral gap condition this implies existence of inertial manifolds in arbitrary space dimensions. The spectral gap condition is satisfied by the Laplace-Beltrami operator for a class of manifolds. The simplest examples are products of sp...

In this paper we explore the connection between semi-classical and quantum Birkhoff canonical forms (BCF) for Schrödinger operators. In particular we give a ”non-symbolic” operator theoretic derivation of the quantum Birkhoff canonical form and provide an explicit recipe for expressing the quantum BCF in terms of the semi-classical BCF.

Let A be a positive (semidefinite) bounded linear operator acting on complex Hilbert space \(\big ({\mathcal {H}}, \langle \cdot , \rangle \big )\). The semi-inner product \({\langle x, y\rangle }_A := Ax, \), \(x, y\in {\mathcal {H}}\) induces seminorm \({\Vert \Vert }_A\) \({\mathcal {H}}\). T an A-bounded {H}}\), the A-numerical radius of is given by $$\begin{aligned} \omega _A(T) = \sup \Bi...