On the Existence and Uniqueness of Ground States of a Generalized Spin-Boson Model

Arai, Asao; Hirokawa, Masao

doi:10.1006/jfan.1997.3140

Cited by 119 publications

(168 citation statements)

References 20 publications

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“…Soft photons are responsible for the absence of a gap in these models. A relatively simple yet interesting model for which the existence (and uniqueness) of the ground state [48,1] as well as gaplessness [28] are known rigorously is the spin-boson Hamiltonian: The model of a two level system coupled to a radiation field. This has also been established for a model of non-relativistic QED [1,6,7,8,9]: A model of nonrelativistic electrons coupled to a radiation field with an ultraviolet cutoff.…”

Section: Fig 1 Crossing Eigenvalues In Born Fock Theorymentioning

confidence: 99%

“…A relatively simple yet interesting model for which the existence (and uniqueness) of the ground state [48,1] as well as gaplessness [28] are known rigorously is the spin-boson Hamiltonian: The model of a two level system coupled to a radiation field. This has also been established for a model of non-relativistic QED [1,6,7,8,9]: A model of nonrelativistic electrons coupled to a radiation field with an ultraviolet cutoff. Unfortunately, for real QED [13,16], where both the electrons and photons are treated as relativistic quantum fields, all that is rigorously known at present is on a perturbative level.…”

Section: Fig 1 Crossing Eigenvalues In Born Fock Theorymentioning

confidence: 99%

“…In order to have explicit error estimates and also in order to make the presentation simple and as elementary as possible, we choose a Gaussian regularizer. 1 , and Φ be the special function defined below:…”

Section: ⊓ ⊔mentioning

confidence: 99%

“…For example, in the case of linearly crossing eigenvalues, the difference in slopes of the eigenvalues at the point of crossing α = (Ė 1 −Ė 2 ), Fig. 1, determines a time scale, 1 α , that takes over as the time scale associated with the gap diverges. An adiabatic theorem that builds on this fact goes back to Born and Fock [15].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

An Adiabatic Theorem without a Gap Condition

Avron¹,

Elgart²

1999

Mathematical Results in Quantum Mechanics

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We prove the adiabatic theorem for quantum evolution without the traditional gap condition. All that this adiabatic theorem needs is a (piecewise) twice differentiable finite dimensional spectral projection. The result implies that the adiabatic theorem holds for the ground state of atoms in quantized radiation field. The general result we prove gives no information on the rate at which the adiabatic limit is approached. With additional spectral information one can also estimate this rate.

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Section: Fig 1 Crossing Eigenvalues In Born Fock Theorymentioning

confidence: 99%

Section: Fig 1 Crossing Eigenvalues In Born Fock Theorymentioning

confidence: 99%

Section: ⊓ ⊔mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

An Adiabatic Theorem without a Gap Condition

Avron¹,

Elgart²

1999

Mathematical Results in Quantum Mechanics

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show abstract

“…Noting A (τ ) κΛ Ω = 0, in the same way as the proof of [3,Lemma 4.7] we estimate (i) and (ii). Concerning (i), one has by (3.4) and (3.5)…”

Section: Overlap With Atomic Ground Statementioning

confidence: 90%

Ground state for point particles interacting through a massless scalar Bose field

Hirokawa

Hiroshima

Spohn

2004

Advances in Mathematics

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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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On the Existence and Uniqueness of Ground States of a Generalized Spin-Boson Model

Cited by 119 publications

References 20 publications

An Adiabatic Theorem without a Gap Condition

An Adiabatic Theorem without a Gap Condition

Ground state for point particles interacting through a massless scalar Bose field

The dressed nonrelativistic electron in a magnetic field

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