Spectral analysis of an abstract pair interaction model

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 $\lambda_{{\rm c},0}$.