Asymptotic Completeness for Rayleigh Scattering

Frölich, J.; Griesemer, Marcel; Schlein, Benjamin

doi:10.1007/s00023-002-8614-9

Cited by 60 publications

(98 citation statements)

References 13 publications

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“…This result is not surprising but the complete proof is long. Actually it follows by mimicking [20,21,16] (see also [34]) and, here, we only give a sketch of the proof. In this appendix we are faced with the lack of smoothness of the ǫ µ (k)'s which define vector fields on spheres |k| =cst (see [33,23]).…”

Section: A Exitences Of a Ground State For The Hamiltonian With Inframentioning

confidence: 99%

The dressed nonrelativistic electron in a magnetic field

Amour

Grébert

Guillot

2006

Math Methods in App Sciences

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We consider a nonrelativistic electron interacting with a classical magnetic field pointing along the x3-axis and with a quantized electromagnetic field. When the interaction between the electron and photons is turned off, the electronic system is assumed to have a ground state of finite multiplicity. Because of the translation invariance along the x3-axis, we consider the reduced Hamiltonian associated with the total momentum along the x3-axis and, after introducing an ultraviolet cutoff and an infrared regularization, we prove that the reduced Hamiltonian has a ground state if the coupling constant and the total momentum along the x3-axis are sufficiently small. We determine the absolutely continuous spectrum of the reduced Hamiltonian and, when the ground state is simple, we prove that the renormalized mass of the dressed electron is greater than or equal to its bare one. We then deduce that the anomalous magnetic moment of the dressed electron is non negative.

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Section: A Exitences Of a Ground State For The Hamiltonian With Inframentioning

confidence: 99%

The dressed nonrelativistic electron in a magnetic field

Amour

Grébert

Guillot

2006

Math Methods in App Sciences

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“…Thus, it is enough to prove that the right hand side converges to the right hand side of (II.17), as tends to 0. This can be shown by adapting arguments in [8] (for Nelson's model) to the present model. We note that…”

Section: Ii2 Virial Theoremmentioning

confidence: 84%

On the Absence of Excited Eigenstates of Atoms in QED

Fröhlich

Pizzo

2009

Commun. Math. Phys.

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For the standard model of QED with static nuclei, nonrelativistic electrons and an ultraviolet cutoff, a new simple proof of absence of excited eigenstates with energies above the groundstate energy and below the ionization threshold of an atom is presented. Our proof is based on a multi-scale virial argument and exploits the fact that, in perturbation theory, excited atomic states decay by emission of one or two photons. Our arguments do not require an infrared cutoff (or regularization) and are applicable for all energies above the groundstate energy, except in a small (α-dependent) interval around the ionization threshold. * also at IHES, Bures-sur-Yvette We begin by summarizing some earlier results closely related to those proven in the present paper. 1) Outside some O(g 2 )-neighborhoods (where g is a coupling constant) of the groundstate energy and of the ionization threshold, proofs for the absence of point spectrum and absolute continuity of the energy spectrum have been given in [2] in the presence of an infrared regularization of the form factor in the interaction coupling the electrons to the quantized radiation field. These proofs are based on an operator renormalization group analysis of the spectrum of a dilated Hamiltonian. 2) An approach involving positive commutators [4], as well as a refined version [3] of the complex spectral deformation method yield analogous results, but without any infrared regularization. The results described in 1) and 2), however, apply only to situations where the decay of an excited state takes place as a consequence of a dipole transition. Hence the assumptions in [2], [3], and [4] do, in general, not cover the entire interval between the groundstate energy and the ionization threshold. For example, the decay of the 2s level of the hydrogen atom, which is due to two-photon or multi-photon transitions, is not understood in these references. 3) Absence of eigenvalues and absolute continuity of the energy spectrum in a neighborhood of the groundstate energy has recently been proven in [9] FP-Abs. Excit. Eigenst., 18-April-2007 2 without any infrared regularization, using a multi-scale version of Mourre's theory.The purpose of this paper is twofold. First, we prove a complete result for the absence of point spectrum below the (unperturbed) ionization threshold and above the groundstate energy, at least for the hydrogen atom. Moreover, and more importantly, we present a proof closely related to the usual physical arguments of perturbation theory, which are made mathematically precise in this paper. Our starting point is the spectroscopic evidence that one-and two-photon transitions are responsible for the decay of all excited atomic or molecular bound states. In fact, starting from a perturbative expansion for a putative excited eigenstate of the total Hamiltonian, we arrive at a contradiction by taking scalar products of the putative eigenvector with trial vectors. These trial vectors represent the decay products of the putative excited states after emission of on...

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“…Hence the asymptotic operators a # + (h) exist on a dense subspace of H. The main results in [17,18,16] apply to both electron-Hamiltonian, (1) and (28).…”

Section: Hence Supmentioning

confidence: 98%

“…The papers [24,25,17,18] Quantitative predictions will be of limited accuracy when relativity or high-energy photons play a significant role.…”

Section: Physical Phenomena and Mathematical Descriptionmentioning

confidence: 99%

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Non-Relativistic Matter and Quantized Radiation

Griesemer

Large Coulomb Systems

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This is a didactic review of spectral and dynamical properties of atoms and molecules at energies below the ionization threshold, the focus being on recent work in which the author was involved. As far as possible, the results are described using a simple model with one electron only, and with scalar bosons. The main ideas are explained but no complete proofs are given. The full-fledged standard model of non-relativistic QED and various of its aspects are described in the appendix.

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Asymptotic Completeness for Rayleigh Scattering

Cited by 60 publications

References 13 publications

The dressed nonrelativistic electron in a magnetic field

The dressed nonrelativistic electron in a magnetic field

On the Absence of Excited Eigenstates of Atoms in QED

Non-Relativistic Matter and Quantized Radiation

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