On the Absence of Eigenvectors of Hamiltonians in a Class of Massless Quantum Field Models without Infrared Cutoff

Arai, Asao; Hirokawa, Masao; Hiroshima, Fumio

doi:10.1006/jfan.1999.3472

Cited by 54 publications

(46 citation statements)

References 28 publications

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“…. , J) is called the infrared regular condition in the GSB model (if this condition is not satisfied for some j, then H GSB may have no ground states [7]). Condition (5.38) with N ≥ 3 means that g j is more infrared regular than the case N = 2.…”

Section: Remark 516mentioning

confidence: 99%

A New Asymptotic Perturbation Theory with Applications to Models of Massless Quantum Fields

Arai

2013

Ann. Henri Poincaré

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Let H 0 and H I be a self-adjoint and a symmetric operator on a complex Hilbert space, respectively, and suppose that H 0 is bounded below and the infimum E 0 of the spectrum of H 0 is a simple eigenvalue of H 0 which is not necessarily isolated. In this paper, we present a new asymptotic perturbation theory for an eigenvalueThe point of the theory is in that it covers also the case where E 0 is a non-isolated eigenvalue of H 0 . Under a suitable set of assumptions, we derive an asymptotic expansion of E(λ) up to an arbitrary finite order of λ as λ → 0. We apply the abstract results to a model of massless quantum fields, called the generalized spinboson model (A. Arai and M. Hirokawa, J. Funct. Anal. 151 (1997), 455-503) and show that the ground state energy of the model has asymptotic expansions in the coupling constant λ as λ → 0.

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Section: Remark 516mentioning

confidence: 99%

A New Asymptotic Perturbation Theory with Applications to Models of Massless Quantum Fields

Arai

2013

Ann. Henri Poincaré

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“…Thus, the present paper looks at the problem from a different angle. Following the physical observation stated below, we adopt an operator-theoretical method in which we combine the technique of spatial localization presented by Griesemer, Lieb, and Loss [14] and an approach based on the proof of the absence of ground state by Arai, Hiroshima, and the author [6]. We believe that this approach is new.…”

Section: §1 Introductionmentioning

confidence: 99%

“…It is worthy of note that Pizzo developed Fröhlich's work in [27]. We tackled IR problem of proving the non-existence of ground state for the so-called generalized spin-boson (GSB) model from an operator-theoretical point of view in [6], while we studied a mathematical mechanism of existence of ground states for it in [4]. However, because GSB model is very general, the information on IR problem for it was so limited that we could not entirely achieve our goal.…”

Section: §1 Introductionmentioning

confidence: 99%

Infrared Catastrophe for Nelson’s Model—Non-Existence of Ground State and Soft-Boson Divergence

Hirokawa¹

2006

Publ. Res. Inst. Math. Sci.

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We mathematically study the infrared catastrophe for the Hamiltonian of Nelson's model when it has the external potential in a general class. For the model, we prove the pull-through formula on ground states in operator theory first. Based on this formula, we show both non-existence of any ground state and divergence of the total number of soft bosons. §1. IntroductionThe purpose of the present paper is to investigate mathematically the infrared (IR) catastrophe for Nelson's Hamiltonian [25], in particular nonexistence of ground state and the divergence of the total number of soft bosons (soft-boson divergence). The exact definition of ground state will be stated in §2. The definition of soft boson will be explained later. IR catastrophe is the trouble of IR divergence caused by massless particles forming a quantized field. Nelson's Hamiltonian is the Hamiltonian of the so-called Nelson's model describing a system of a quantum particle, which moves in the 3-dimensional Euclidean space R 3 under the influence of an external potential, and which interacts with a massless scalar Bose field. The massless scalar Bose field is the quantized scalar field made of massless bosons. The boson is the (quantum)

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“…It turns out that it is a very subtle problem whether or not ground states exist in the original Hilbert space where the unperturbed part of the Hamiltonian of the model is defined via the usual Fock representation of canonical commutation relations. This aspect was investigated in [16] in view of absence of ground states.…”

Section: Introductionmentioning

confidence: 99%

“…In contrast to the paper [16], we analyze, in the present paper, mathematical structures for existence (in the original Hilbert space) of ground states of the massless model without infrared regularity condition (for the exact meaning, see the paragraph after Remark 1.2 below). We establish existence theorems of ground states in the massless theory in terms of quantities defined as the mass-zero limits of those in the massive (regularized) theory.…”

Section: Introductionmentioning

confidence: 99%

Ground States of a General Class of Quantum Field Hamiltonians

Arai

Hirokawa

2000

Rev. Math. Phys.

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We consider a model of a quantum mechanical system coupled to a (massless) Bose field, called the generalized spin-boson model (A. Arai and M. Hirokawa, J. Funct. Anal. 151 (1997), 455-503), without infrared regularity condition. We define a regularized Hamiltonian H(ν) with a parameter ν ≥ 0 such that H = H(0) is the Hamiltonian of the original model. We clarify a relation between ground states of H(ν) and those of H by formulating sufficient conditions under which weak limits, as ν → 0, of the ground states of H(ν)'s are those of H. We also establish existence theorems on ground states of H(ν) and H under weaker conditions than in the previous paper mentioned above.Mathematics Subject Classifications (1991): 81Q10, 47B25, 47N50

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On the Absence of Eigenvectors of Hamiltonians in a Class of Massless Quantum Field Models without Infrared Cutoff

Cited by 54 publications

References 28 publications

A New Asymptotic Perturbation Theory with Applications to Models of Massless Quantum Fields

A New Asymptotic Perturbation Theory with Applications to Models of Massless Quantum Fields

Infrared Catastrophe for Nelson’s Model—Non-Existence of Ground State and Soft-Boson Divergence

Ground States of a General Class of Quantum Field Hamiltonians

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