Localization of two-dimensional massless Dirac fermions in a magnetic quantum dot

We consider a two-dimensional massless Dirac operator $H$ in the presence of a perturbed homogeneous magnetic field $B=B_0+b$ and a scalar electric potential $V$. For $V\in L_{\rm loc}^p(\R^2)$, $p\in(2,\infty]$, and $b\in L_{\rm loc}^q(\R^2)$, $q\in(1,\infty]$, both decaying at infinity, we show that states in the discrete spectrum of $H$ are superexponentially localized. We establish the existence of such states between the zeroth and the first Landau level assuming that V=0. In addition, under the condition that $b$ is rotationally symmetric and that $V$ satisfies certain analyticity condition on the angular variable, we show that states belonging to the discrete spectrum of $H$ are Gaussian-like localized

Ground states of semi-relativistic Pauli-Fierz and no-pair Hamiltonians in QED at critical Coulomb coupling

We consider the semi-relativistic Pauli-Fierz Hamiltonian and a no-pair model of a hydrogen-like atom interacting with a quantized photon field at the respective critical values of the Coulomb coupling constant. For arbitrary values of the fine-structure constant and the ultra-violet cutoff, we prove the existence of normalizable ground states of the atomic system in both models. This complements earlier results on the existence of ground states in (semi-)relativistic models of quantum electrodynamics at sub-critical coupling by E. Stockmeyer and the present authors. Technically, the main new achievement is an improved estimate on the spatial exponential localization of low-lying spectral subspaces yielding uniform bounds at large Coulomb coupling constants. In the semi-relativistic Pauli-Fierz model our exponential decay rate given in terms of the binding energy reduces to the one known from the electronic model when the radiation field is turned off. In particular, an increase of the binding energy due to the radiation field is shown to improve the localization of ground states.Comment: Revised second version providing optimized exponential decay rates for the semi-relativistic Pauli-Fierz model; see Theorem 4.5 and Remark 4.7. The revised manuscript is accepted for publication in the Journal of Operator Theory. 25 page

The mass shell in the semi-relativistic Pauli-Fierz model

We consider the semi-relativistic Pauli-Fierz model for a single free electron interacting with the quantized radiation field. Employing a variant of Pizzo's iterative analytic perturbation theory we construct a sequence of ground state eigenprojections of infra-red cutoff, dressing transformed fiber Hamiltonians and prove its convergence, as the cutoff goes to zero. Its limit is the ground state eigenprojection of a certain Hamiltonian unitarily equivalent to a renormalized fiber Hamiltonian acting in a coherent state representation space. The ground state energy is an exactly two-fold degenerate eigenvalue of the renormalized Hamiltonian, while it is not an eigenvalue of the original fiber Hamiltonian unless the total momentum is zero. These results hold true, for total momenta inside a ball about zero of arbitrary radius p>0, provided that the coupling constant is sufficiently small depending on p and the ultra-violet cutoff. Along the way we prove twice continuous differentiability and strict convexity of the ground state energy as a function of the total momentum inside that ball.Comment: 44 page

An In nite Level Atom coupled to a Heat Bath Dissertation zur Erlangung des Grades "Doktor der Naturwissenschaften" am Fachbereich Physik, Mathematik und Informatik

We study the mathematics of a nite particle system coupled to a heat bath. The Standard Model of Quantum Electrodynamics at temperature zero yields a Hamiltonian H describing the energy of particles interacting with photons. In the Heisenberg picture the time evolution of the physical system is the action of a one-parameter-group (τt)t∈R on a set of observables A: τt: A ↦ → τt(A), t ∈ R, A ∈ A Note, that τ is related with solutions of the Schrödinger equation for H. To consider states of A describing the physical system near its thermal equilibrium at temperature T> 0, we follow the ansatz of Jaksic and Pillet to construct a representation of A. Now, states are unit vectors in this representation and the time evolution, is described with the aid of the Standard Liouvillean L. The following results are derived or proved, respectively, in this thesis:- the construction of the representation- the self-adjointness of the Standard Liouvillean- the existence of an equilibrium state in the representation- the limit of large times for the physical system.