We consider an abstract pairinteraction 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 $\lambda_{{\rm c},0}$.
MSC(Primary)  47N50; 

MSC(Secondary)  47B25; 81T10;


Uncontrolled Keywords  quantum field;
pairinteraction model;
spectral analysis;
Bogoliubov transformation;

