Spectral Theory of Massless Pauli-Fierz Models

Georgescu, Vladimir; Gérard, Christian; Møller, Jacob Schach

doi:10.1007/s00220-004-1111-x

Cited by 61 publications

(161 citation statements)

References 22 publications

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“…The drawback ofB is that it is only symmetric, but not self-adjoint, and hence not admissible as a conjugate operator. Therefore Skibsted, and, later, Georgescu, Gérard, and Møller developed suitable extensions of conjugate operator theory that allow for non-selfadjoint conjugate operators [24,13]. Skibsted applied his conjugate operator theory to (1) and obtained absolute continuity of the energy spectrum away from thresholds and eigenvalues under an infrared (IR) regularization, but not for (3).…”

Section: Introductionmentioning

confidence: 99%

“…Hübner and Spohn study the spin-boson model with massive bosons or with a photon number cutoff imposed. Their work inspired [24] and [13], where better results were obtained: Skibsted analyzed (1) and assumed that |κ(k)| ≤ |k| 5/2 , while, in [13], |κ(k)| ≤ |k| β , with β > 1/2, is sufficient for a Nelson-type model with scalar bosons. The main achievement of [13] is that no bound on the coupling strength is required.…”

Section: Introductionmentioning

confidence: 99%

“…Their work inspired [24] and [13], where better results were obtained: Skibsted analyzed (1) and assumed that |κ(k)| ≤ |k| 5/2 , while, in [13], |κ(k)| ≤ |k| β , with β > 1/2, is sufficient for a Nelson-type model with scalar bosons. The main achievement of [13] is that no bound on the coupling strength is required. Papers [6] and [7] do not introduce an infrared regularization but establish the spectral properties mentioned above only away from O(α 3 )-neighborhoods of the particle ground state energy and the ionization threshold.…”

Section: Introductionmentioning

confidence: 99%

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Spectral Theory for the Standard Model of Non-Relativistic QED

Fröhlich

Griesemer

Sigal

2008

Commun. Math. Phys.

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For a model of atoms and molecules made from static nuclei and non-relativistic electrons coupled to the quantized radiation field (the standard model of non-relativistic QED), we prove a Mourre estimate and a limiting absorption principle in a neighborhood of the ground state energy. As corollaries we derive local decay estimates for the photon dynamics, and we prove absence of (excited) eigenvalues and absolute continuity of the energy spectrum near the ground state energy, a region of the spectrum not understood in previous investigations.The conjugate operator in our Mourre estimate is the second quantized generator of dilatations on Fock space.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Spectral Theory for the Standard Model of Non-Relativistic QED

Fröhlich

Griesemer

Sigal

2008

Commun. Math. Phys.

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“…There is a large body of literature on this subject, dealing with different variants of the standard model of nonrelativistic QED; see e.g. [2], [3], [4], [6], [7], [10], [9], [14], [18]. For a review of previous work, we also refer the reader to the Introduction of [9].…”

Section: Description Of the Problem And Summary Of Main Resultsmentioning

confidence: 99%

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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“…[BCHM, CGH, DJ, GGo] par exemple) et de nouveaux développements apparaissent pour traiter des difficultés particulières en théorie de champs (cf. [GGM1,GGM2]). …”

Section: Introductionunclassified

Un nouveau regard sur l’estimation de Mourre

Golenia¹

2008

Journées Équations Aux Dérivées Partielles

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Spectral Theory of Massless Pauli-Fierz Models

Cited by 61 publications

References 22 publications

Spectral Theory for the Standard Model of Non-Relativistic QED

Spectral Theory for the Standard Model of Non-Relativistic QED

On the Absence of Excited Eigenstates of Atoms in QED

Un nouveau regard sur l’estimation de Mourre

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