Spectral Theory of Pauli–Fierz Operators

Dereziński, Jan; Jakšić, Vojkan

doi:10.1006/jfan.2000.3681

Cited by 98 publications

(132 citation statements)

References 44 publications

(37 reference statements)

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“…The "left" reservoir consists of the particles with labels in (−∞, −N ] and the right reservoir consists of the particles with labels in [N, +∞). The system a A series of results has been obtained for systems coupled to a single reservoir, such as nonrelativistic atoms coupled to the quantized electromagnetic field at temperature zero 3,4,5 , finite level atoms coupled to a boson field at positive temperature 43,44,19 , classical particles coupled to a scalar field at temperature zero 49,50 or at positive temperature 45 . b For quantum spin systems axioms are formulated in [78] which establish the existence of a stationary state and its mixing property for finite systems coupled to several reservoirs at different temperatures luc10: submitted to World Scientific on February 5, 2008consists of the particles in the middle.…”

Section: Hamiltonian Reservoirsmentioning

confidence: 99%

Fourier's Law: A Challenge to Theorists

Bonetto

Lebowitz

Rey-Bellet

2000

Mathematical Physics 2000

354

525

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We present a selective overview of the current state of our knowledge (more precisely of our ignorance) regarding the derivation of Fourier's Law, J(r) = −κ∇T (r); J the heat flux, T the temperature and κ, the heat conductivity. This law is empirically well tested for both fluids and crystals, when the temperature varies slowly on the microscopic scale, with κ an intrinsic property which depends only on the system's equilibrium parameters, such as the local temperature and density. There is however at present no rigorous mathematical derivation of Fourier's law and ipso facto of Kubo's formula for κ, involving integrals over equilibrium time correlations, for any system (or model) with a deterministic, e.g. Hamiltonian, microscopic evolution.

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Section: Hamiltonian Reservoirsmentioning

confidence: 99%

Fourier's Law: A Challenge to Theorists

Bonetto

Lebowitz

Rey-Bellet

2000

Mathematical Physics 2000

354

525

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“…Here η σ , η σ ∈ C ∞ (R 3 ) is a partition of unity, η 2 σ + (η σ ) 2 = 1, with η σ (k) = 1 for |k| ≤ 2σ and η σ (k) = 1 for |k| ≥ 4σ. It follows that B σ = B σ ⊗ 1 with respect to H = H σ ⊗ F σ , and that (9) and the virial theorem, P σ [H σ , B σ ]P σ = 0, imply that…”

Section: Introductionmentioning

confidence: 99%

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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“…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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Spectral Theory of Pauli–Fierz Operators

Cited by 98 publications

References 44 publications

Fourier's Law: A Challenge to Theorists

Fourier's Law: A Challenge to Theorists

Spectral Theory for the Standard Model of Non-Relativistic QED

On the Absence of Excited Eigenstates of Atoms in QED

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