The excited states of a charged particle interacting with the quantized electromagnetic field and an external potential all decay, but such a particle should have a true ground state -one that minimizes the energy and satisfies the Schrödinger equation. We prove quite generally that this state exists for all values of the fine-structure constant and ultraviolet cutoff. We also show the same thing for a many-particle system under physically natural conditions. c 2000 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes. MGELML/3/2/01 2 exists (as far as we believe now) is the free particle (i.e., particle plus field). In the presence of an external potential, however, like the Coulomb potential of a nucleus, a ground state should exist.The difficulty in establishing this ground state comes from the fact that the bottom of the spectrum lies in the continuum (i.e., essential spectrum), not below it, as is the case for the usual Schrödinger equation. We denote the bottom of spectrum of the free-particle Hamiltonian for N particles with appropriate statistics by E 0 (N ). The 'free-particle' Hamiltonian includes the interparticle interaction (e.g., the Coulomb repulsion of electrons) but it does not include the interaction with a fixed external potential, e.g., the interaction with nuclei. When the latter is included we denote the bottom of the spectrum by E V (N ). It is not hard to see in many cases thatis, nevertheless, the bottom of the essential spectrum. The reason is that we can always add arbitrarily many, arbitrarily 'soft' photons that add arbitrarily little energy. It is the soft photon problem that is our primary concern here. The main point of this paper is to show how to overcome this infrared problem and to show, quite generally for a one-particle system, that a ground state exists for all values of the particle mass, the coupling to the field (fine-structure constant α = e 2 / c), the magnetic g-factor, and the ultraviolet cutoff Λ of the electromagnetic field frequencies, provided a bound state exists when the field is turned off. This result implies, in particular, that for a fixed ultraviolet cutoff renormalization of the various physical quantities will not affect the existence of a ground state. Of course, nothing can be said about the limit as the cutoff tends to infinity. We also include a large class of interactions much more general than the usual Coulomb interaction.The model we discuss has been used quite frequently in field theory. In its classical version it was investigated by Kramers [12] who seems to have been the first to point out the possibility of renormalization. The quantized version was investigated by Pauli and Fierz [24] in connection with scattering theory. Most importantly, it was used by Bethe [9] to obtain a suprisingly good value for the Lamb shift.Various restricted versions of the problem have been attacked successfully. In the early seventies Fröhlich investigated the infrared problem in translation invariant models of scalar elect...

Scattering in a model of a massive quantum-mechanical particle, an "electron", interacting with massless, relativistic bosons, "photons", is studied. The interaction term in the Hamiltonian of our model describes emission and absorption of "photons" by the "electron"; but "electron-positron" pair production is suppressed. An ultraviolet cutoff and an (arbitrarily small, but fixed) infrared cutoff are imposed on the interaction term. In a range of energies where the propagation speed of the dressed "electron" is strictly smaller than the speed of light, unitarity of the scattering matrix is proven, provided the coupling constant is small enough; (asymptotic completeness of Compton scattering). The proof combines a construction of dressed one-electron states with propagation estimates for the "electron" and the "photons".

It is expected that the state of an atom or molecule, initially put into an excited state with an energy below the ionization threshold, relaxes to a groundstate by spontaneous emission of photons which propagate to spatial infinity. In this paper, this picture is established for a large class of models of non-relativistic atoms and molecules coupled to the quantized radiation field, but with the simplifying feature that an (arbitrarily tiny, but positive) infrared cutoff is imposed on the interaction Hamiltonian.This result relies on a proof of asymptotic completeness for Rayleigh scattering of light on an atom. We establish asymptotic completeness of Rayleigh scattering for a class of model Hamiltonians with the features that the atomic Hamiltonian has point spectrum coexisting with absolutely continuous spectrum, and that either an infrared cutoff is imposed on the interaction Hamiltonian or photons are treated as massive particles.We show that, for models of massless photons, the spectrum of the Hamiltonian strictly below the ionization threshold is purely continuous, except for the groundstate energy.

In models of (non-relativistic and pseudo-relativistic) electrons interacting with static nuclei and with the (ultraviolet-cutoff) quantized radiation field, the existence of asymptotic electromagnetic fields is established. Our results yield some mathematically rigorous understanding of Rayleigh scattering and of the phenomenon of relaxation of isolated atoms to their ground states. Our proofs are based on propagation estimates for electrons inspired by similar estimates known from N-body scattering theory.

A minimax 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$ dinger operator with the same potential.