Absence of Ground States in the Translation Invariant Massless Nelson Model

2020 ◽  
Vol 21 (8) ◽  
pp. 2655-2679
Author(s):  
Thomas Norman Dam
Keyword(s):  
Ground States ◽  
Translation Invariant ◽  
Nelson Model

scholarly journals Coulomb scattering in the massless Nelson model II. Regularity of ground states

2019 ◽  
Vol 31 (03) ◽  
pp. 1950010 ◽  
Author(s):  
Wojciech Dybalski ◽  
Alessandro Pizzo
Keyword(s):  
Perturbation Theory ◽  
Electron Scattering ◽  
Scattering Theory ◽  
Ground States ◽  
Total Momentum ◽  
Coulomb Scattering ◽  
Coupling Constant ◽  
Analytic Perturbation Theory ◽  
Nelson Model ◽  
Scattering States

For the massless Nelson model, we provide detailed information about the dependence of the normalized ground states [Formula: see text] of the fiber single-electron Hamiltonians [Formula: see text] on the total momentum [Formula: see text] and the infrared cut-off [Formula: see text]. This information is obtained with the help of the iterative analytic perturbation theory. In particular, we derive bounds of the form [Formula: see text] for some constant [Formula: see text] and a function of the maximal admissible coupling constant [Formula: see text] such that [Formula: see text]. These results hold both in the infrared-regular and infrared-singular cases. They are exploited in part I of this series to construct the two-electron scattering states in the infrared-regular massless Nelson model (in the absence of an infrared cut-off) along the lines of the Haag–Ruelle scattering theory. They should also be relevant to the problem of scattering of two infraparticles in the infrared-singular Nelson model, whose solution is the goal of this series of papers. Although a part of a larger investigation, the present work is written in a self-contained fashion.

THE POLARON REVISITED

2006 ◽  
Vol 18 (05) ◽  
pp. 485-517 ◽  
Author(s):  
JACOB SCHACH MØLLER
Keyword(s):  
Dispersion Relation ◽  
Large Class ◽  
Spectral Properties ◽  
Phonon Dispersion ◽  
Total Momentum ◽  
Translation Invariant ◽  
Polaron Model ◽  
Nelson Model ◽  
Phonon Dispersion Relation

In recent years, the spectral properties of the translation invariant Nelson model has been studied. Some of the results obtained did not extend to the related polaron model for technical reasons related to the typical assumption of boundedness of the phonon dispersion relation in the polaron model. In this paper, we work with a large class of linearly coupled translation invariant models which includes both the Nelson model and H. Fröhlich's polaron model. The problems considered are chosen based on relevance for the polaron model. A key input is an analysis of the behavior of the bottom of the spectrum of the fiber Hamiltonians at large total momentum.

The Calder´on-Zygmund Theory I: Ellipticity

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
Classical Theory ◽  
Singular Integral ◽  
Singular Integrals ◽  
Differential Operators ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Single Parameter ◽  
Partial Differential ◽  
Translation Invariant ◽  
Partial Differential Operators

This chapter discusses a case for single-parameter singular integral operators, where ρ‎ is the usual distance on ℝn. There, we obtain the most classical theory of singular integrals, which is useful for studying elliptic partial differential operators. The chapter defines singular integral operators in three equivalent ways. This trichotomy can be seen three times, in increasing generality: Theorems 1.1.23, 1.1.26, and 1.2.10. This trichotomy is developed even when the operators are not translation invariant (many authors discuss such ideas only for translation invariant, or nearly translation invariant operators). It also presents these ideas in a slightly different way than is usual, which helps to motivate later results and definitions.

Ground states for the ising model with an external field on the Cayley tree

2018 ◽  
Vol 2018 (3) ◽  
pp. 147-155
Author(s):  
M.M. Rakhmatullaev ◽  
M.A. Rasulova
Keyword(s):  
Ising Model ◽  
External Field ◽  
Cayley Tree ◽  
Ground States
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