Marcel Griesemer

A Minimax Principle for the Eigenvalues in Spectral Gaps

Aminimax principle is derived for the eigenvalues in the spectral gap of a possibly non-semibounded selfadjoint operator. It allows the nth eigenvalue of the Dirac operator with Coulomb potential from below to be bound by the nth eigenvalue of a semibounded Hamiltonian which is of interest in the context of stability of matter. As a second application it is shown that the Dirac operator with suitable non-positive potential has at least as many discrete eigenvalues as the Schro

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

dinger operator with the same potential.

ON SPECTRAL RENORMALIZATION GROUP

Abstract.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 α3/2, α being the fine structure constant. A suitably chosen ground state vector depends analytically on α3/2 and it is twice continuously differentiable with respect to the nuclear coordinates.

On the Magnetic Pekar Functional and the Existence of Bipolarons

The operator-theoretic renormalization group (RG) methods are powerful analytic tools to explore spectral properties of field-theoretical models such as quantum electrodynamics (QED) with non-relativistic matter. In this paper, these methods are extended and simplified. In a companion paper, our variant of operator-theoretic RG methods is applied to establishing the limiting absorption principle in non-relativistic QED near the ground state energy.

SPECTRAL RENORMALIZATION GROUP AND LOCAL DECAY IN THE STANDARD MODEL OF NON-RELATIVISTIC QUANTUM ELECTRODYNAMICS

First, this paper proves the existence of a minimizer for the Pekar functional including a constant magnetic field and possibly some additional local fields that are energy reducing. Second, the existence of the aforementioned minimizer is used to establish the binding of polarons in the model of Pekar–Tomasevich including external fields.

Instability of a pseudo-relativistic model of matter with self-generated magnetic field

We prove a limiting absorption principle for the standard model of non-relativistic quantum electrodynamics (QED) and for Nelsons model describing interactions of electrons with phonons. To this end, we use the spectral renormalization group technique on the continuous spectrum in conjunction with Mourre theory.

On the Atomic Photoeffect in Non-relativistic QED

For a pseudo-relativistic model of matter, based on the no-pair Hamiltonian, we prove that the inclusion of the interaction with the self-generated magnetic field leads to instability for all positive values of the fine structure constant. This is true no matter whether this interaction is accounted for by the Breit potential, by an external magnetic field which is chosen to minimize the energy, or by the quantized radiation field.

Self-adjointness and domain of the Fröhlich Hamiltonian

In this paper we present a mathematical analysis of the photoelectric effect for one-electron atoms in the framework of non-relativistic QED. We treat photo-ionization as a scattering process where in the remote past an atom in its ground state is targeted by one or several photons, while in the distant future the atom is ionized and the electron escapes to spacial infinity. Our main result shows that the ionization probability, to leading order in the fine-structure constant, α, is correctly given by formal time-dependent perturbation theory, and, moreover, that the dipole approximation produces an error of only sub-leading order in α. In this sense, the dipole approximation is rigorously justified.

The strong-coupling polaron in static electric and magnetic fields

In the large polaron model of Herbert Frohlich, the electron-phonon interaction is a small perturbation in form sense, but a large perturbation in operator sense. This means that the form-domain of the Hamiltonian is not affected by the interaction but the domain of self-adjointness is. In the particular case of the Frohlich model, we are nevertheless able, thanks to a recently published new operator bound, to give an explicit characterization of the domain in terms of a suitable dressing transform. Using the mapping properties of this dressing transform, we analyse the smoothness of vectors in the domain of the Hamiltonian with respect to the position of the electron.

Non-Relativistic Matter and Quantized Radiation

This paper is concerned with Frohlich polarons subject to static electric and magnetic fields in the limit of large electron–phonon coupling. To leading order in the coupling constant, , the ground state energy is shown to be correctly given by the minimum of the Pekar functional including the electromagnetic fields, provided these fields in the Frohlich model are scaled properly with α. As a corollary, the binding of two polarons in strong magnetic fields is obtained.