Ground State Properties of the Nelson Hamiltonian: A Gibbs Measure-Based Approach

Betz, Volker; Hiroshima, Fumio; Lőrinczi, József; Minlos, R. A.; Spohn, Herbert

doi:10.1142/s0129055x02001119

Cited by 48 publications

(53 citation statements)

References 19 publications

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“…See, apart from [14] mentioned above, the papers [1,3,21]. As for the massless confined model we refer the reader to [7,9,24,26,38].…”

Section: Non-relativistic Qed: An Overviewmentioning

confidence: 99%

The Translation Invariant Massive Nelson Model: I. The Bottom of the Spectrum

Møller

2005

Ann. Henri Poincaré

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In this paper we analyze the bottom of the energy-momentum spectrum of the translation invariant Nelson model, describing one electron linearly coupled to a second quantized massive scalar field. Our results are valid for all values of the coupling constant and include an HVZ theorem, non-degeneracy of ground states, existence of isolated groundstates in dimensions 1 and 2, non-existence of ground states embedded in the bottom of the essential spectrum in dimensions 3 and 4, (i.e., at total momenta where no isolated groundstate eigenvalue exists), and we study regularity and monotonicity properties of the bottom of the essential spectrum, as a function of total momentum.

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“…See, apart from [14] mentioned above, the papers [1,3,21]. As for the massless confined model we refer the reader to [7,9,24,26,38].…”

Section: Non-relativistic Qed: An Overviewmentioning

confidence: 99%

The Translation Invariant Massive Nelson Model: I. The Bottom of the Spectrum

Møller

2005

Ann. Henri Poincaré

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“…Arai et al [3,4] and Betz et al [11] studied the absence of ground states of some model and Dereziński and Gérard [18], Hirokawa [23], and Lőrinczi [34] established that of the Nelson model. However, Arai [1] takes a non-Fock representation and then shows the existence of a ground state of H under (1.8).…”

Section: The Nelson Modelmentioning

confidence: 99%

Enhanced binding of an N-particle system interacting with a scalar bose field I

Hiroshima

Sasaki

2007

Math. Z.

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“…In the present paper the soft boson means the boson in a ground state. Recently, the spectral properties of Nelson's Hamiltonian has been studied rather intensively (e.g., [2,9,11,16,20,24]). In particular, Betz et al showed in [9] that when the external potential is in the Kato class the total number of soft bosons for Nelson's Hamiltonian diverges under the infrared singularity (IRS) condition.…”

Section: §1 Introductionmentioning

confidence: 99%

“…Around the same time Lőrinczi et al showed in [24] that when the external potential is strongly confining there is no ground state of Nelson's Hamiltonian in spatial dimension 3. The results in both [9] and [24] are proved by means of functional integrals.…”

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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Ground State Properties of the Nelson Hamiltonian: A Gibbs Measure-Based Approach

Cited by 48 publications

References 19 publications

The Translation Invariant Massive Nelson Model: I. The Bottom of the Spectrum

The Translation Invariant Massive Nelson Model: I. The Bottom of the Spectrum

Enhanced binding of an N-particle system interacting with a scalar bose field I

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

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