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A remark on the binding condition by the decay of particle's potential

Toshimitsu Takaesu

TL;DR

This work analyzes a non-relativistic particle interacting with a scalar Bose field, formulating the system on a product Hilbert space and defining the total Hamiltonian $H(\kappa)$. Under decay assumptions (A.1) and (A.2) on the particle potential $V$, it proves the binding condition $E(H(\kappa)) < \Sigma_{\infty}(H(\kappa))$, without requiring the particle’s ground state as a premise. The main technical flow combines local energy estimates and translation invariance to compare the bound-state energy with the bottom of the essential spectrum, and an immediate implication is the exponential decay of the ground state when it exists, via results such as Gr04. Overall, the paper provides a sufficient condition for binding in particle-field models and derives decay properties of bound states, contributing to the spectral analysis of quantum field systems.

Abstract

The system of a particle interacting with a Bose field is investigated. It is proven that the binding condition holds by the decay of particle's potential. As an application, the exponential decay of the ground state follows.

A remark on the binding condition by the decay of particle's potential

TL;DR

This work analyzes a non-relativistic particle interacting with a scalar Bose field, formulating the system on a product Hilbert space and defining the total Hamiltonian . Under decay assumptions (A.1) and (A.2) on the particle potential , it proves the binding condition , without requiring the particle’s ground state as a premise. The main technical flow combines local energy estimates and translation invariance to compare the bound-state energy with the bottom of the essential spectrum, and an immediate implication is the exponential decay of the ground state when it exists, via results such as Gr04. Overall, the paper provides a sufficient condition for binding in particle-field models and derives decay properties of bound states, contributing to the spectral analysis of quantum field systems.

Abstract

The system of a particle interacting with a Bose field is investigated. It is proven that the binding condition holds by the decay of particle's potential. As an application, the exponential decay of the ground state follows.
Paper Structure (2 sections, 8 theorems, 43 equations)

This paper contains 2 sections, 8 theorems, 43 equations.

Table of Contents

  1. Introduction and Main Result
  2. Proof of Theorem \ref{['gapSigma']}

Key Result

Theorem 1.1

(Binding condition) Assume (A.1) and (A.2). Then,

Theorems & Definitions (11)

  • Remark 1.1
  • Remark 1.2
  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Corollary 2.5
  • Remark 2.1
  • Proposition 2.6
  • ...and 1 more