Existence of ground states of hydrogen-like atoms in relativistic quantum electrodynamics. II. The no-pair operator

Könenberg, Martin; Matte, Oliver; Stockmeyer, Edgardo

doi:10.1063/1.3658863

Cited by 14 publications

(56 citation statements)

References 30 publications

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“…The family of probability measures µ SRPF t , which is our main object in this paper, is defined on the set of cádlág paths, and its pair interaction is not uniformly bounded. We prove that µ SRPF t converges to a probability measure µ SRPF ∞ in the local weak sense as t → ∞ by using the existence of the ground state of H, which is studied in [HH13a, KMS09,KMS11]. This paper is organized as follows: Section 2 is devoted to defining the SRPF Hamiltonian H qf in both a Fock space and a function space to study the semigroup by a path measure.…”

Section: Self-adjoint Extensions and Functional Integrationsmentioning

confidence: 99%

Functional integral approach to semi-relativistic Pauli–Fierz models

Hiroshima

2014

Advances in Mathematics

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By means of functional integrations spectral properties of semi-relativistic Pauli-Fierz Hamiltoniansin quantum electrodynamics is considered. Here p is the momentum operator, A a quantized radiation field on which an ultraviolet cutoff is imposed, V an external potential, H rad the free field Hamiltonian and m ≥ 0 describes the mass of electron. Two self-adjoint extensions of a semi-relativistic Pauli-Fierz Hamiltonian are defined. The Feynman-Kac type formula of e −tH is given. An essential self-adjointness, a spatial decay of bound states, a Gaussian domination of the ground state and the existence of a measure associated with the ground state are shown. All the results are independent of values of coupling constant α, and it is emphasized that m = 0 is included.

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Section: Self-adjoint Extensions and Functional Integrationsmentioning

confidence: 99%

Functional integral approach to semi-relativistic Pauli–Fierz models

Hiroshima

2014

Advances in Mathematics

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“…In particular when (m, M) ∈ [0, ∞) × (0, ∞), one can show the existence of ground state. This is actually done in [KMS11a,KMS11b]. It is emphasized that E m for m = 0 is the edge of the continuous spectrum and there is no positive gap between E m and inf σ(H m ) \ {E m }.…”

Section: Introductionmentioning

confidence: 99%

Spectrum of the semi-relativistic Pauli–Fierz model I

Hidaka¹,

Hiroshima²

2016

Journal of Mathematical Analysis and Applications

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The existence of the ground state of the so-called semi-relativistic Pauli-Fierz model is proven. Let A be a quantized radiation field and H f,m the free field Hamiltonians which is the second quantization of |k| 2 + m 2 . It has been established so far that the semi-relativistic Pauli-Fierz modelhas the unique ground state for (m, M ) ∈ {(0, ∞) × [0, ∞)} ∪ {[0, ∞) × (0, ∞)}. In this paper the existence of the ground state of H SRPF with (m, M ) ∈ [0, ∞) × [0, ∞) is shown. We emphasize that our results include a singular case (m, M ) = (0, 0), i.e., the existence of the ground state of the Hamiltonian of the form: |− i∇ ⊗ 1l − A| + V ⊗ 1l + 1l ⊗ H f,0 is established.

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“…We discuss these higher order estimates in Subsection 6.4 but refrain from repeating their proofs. We remark that many of the arguments presented in Section 6 are alternatives to those used in [28,29]. The second step in the proof of the existence of ground states comprises of a compactness argument showing that every sequence {φ mj } with m j ց 0 contains a strongly convergent subsequence.…”

Section: Introductionmentioning

confidence: 99%

“…In fact, one readily verifies that the limit of such a subsequence is a ground state eigenfunction of the original Hamiltonian with massless photons. This step is performed in Section 8, in parts by means of arguments alternative to those in [28,29]. The compactness argument requires, however, a number of non-trivial ingredients.…”

Section: Introductionmentioning

confidence: 99%

“…We outline the proof of the latter formula and of the soft photon bound for the semi-relativistic Pauli-Fierz operator in Section 7. The photon derivative bound and the infra-red bounds for the no-pair operator are derived by very similar procedures and we refer the interested reader to our original articles [28,29] for the rather dull technical details. The arguments presented in Section 7 are also intended to emphasize the role of the gauge invariance of the models treated here.…”

Section: Introductionmentioning

confidence: 99%

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Hydrogen-Like Atoms in Relativistic Qed

Könenberg

Matte

Stockmeyer

2013

Complex Quantum Systems

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In this review we consider two different models of a hydrogenic atom in a quantized electromagnetic field that treat the electron relativistically. The first one is a no-pair model in the free picture, the second one is given by the semi-relativistic Pauli-Fierz Hamiltonian. For both models we discuss the semi-boundedness of the Hamiltonian, the strict positivity of the ionization energy, and the exponential localization in position space of spectral subspaces corresponding to energies below the ionization threshold. Moreover, we prove the existence of degenerate ground state eigenvalues at the bottom of the spectrum of the Hamiltonian in both models. All these results hold true, for arbitrary values of the finestructure constant, e 2 , and the ultra-violet cut-off, and for a general class of electrostatic potentials including the Coulomb potential with nuclear charges less than (sometimes including) the critical charges without radiation field, namely e −2 2/π for the semi-relativistic Pauli-Fierz operator and e −2 2/(2/π + π/2) for the no-pair operator. Apart from a detailed discussion of diamagnetic inequalities in QED (which are applied to study the semi-boundedness) all results stem from earlier articles written by the authors. While a few proofs are merely sketched, we streamline earlier proofs or present alternative arguments at many places.

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Existence of ground states of hydrogen-like atoms in relativistic quantum electrodynamics. II. The no-pair operator

Cited by 14 publications

References 30 publications

Functional integral approach to semi-relativistic Pauli–Fierz models

Functional integral approach to semi-relativistic Pauli–Fierz models

Spectrum of the semi-relativistic Pauli–Fierz model I

Hydrogen-Like Atoms in Relativistic Qed

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