Spectral analysis of an abstract pair interaction model

TL;DR

This paper investigates the spectral properties of an abstract pair-interaction model in quantum field theory, identifying parameter regimes where the Hamiltonian can be explicitly diagonalized and analyzing the spectrum's behavior.

Contribution

It provides a detailed spectral analysis of the Hamiltonian for both massive and massless cases, including conditions for diagonalization via Bogoliubov transformations.

Findings

Explicit spectrum identification for certain coupling constants.

Demonstration of unboundedness of the Hamiltonian outside specific regimes.

Difference in spectral behavior between massive and massless cases.

Abstract

We consider an abstract pair-interaction model in quantum field theory with a coupling constant $\lambda\in {\mathbb R}$ and analyze the Hamiltonian $H(\lambda)$ of the model. In the massive case, there exist constants $\lambda_{\rm c}<0$ and $\lambda_{{\rm c},0}<\lambda_{\rm c}$ such that, for each $\lambda \in (\lambda_{{\rm c},0},\lambda_{\rm c})\cup (\lambda_{\rm c},\infty)$, $H(\lambda)$ is diagonalized by a proper Bogoliubov transformation, so that the spectrum of $H(\lambda)$ is explicitly identified, where the spectrum of $H(\lambda)$ for $\lambda>\lambda_{\rm c}$ is different from that for $\lambda\in (\lambda_{{\rm c},0}, \lambda_{\rm c})$. As for the case $\lambda<\lambda_{{\rm c},0}$, we show that $H(\lambda)$ is unbounded from above and below. In the massless case, $\lambda_{\rm c}$ coincides with $\lambda_{{\rm c},0}$.