Spectral analysis of an abstract pair interaction model

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 ${\lambda }_{\mathrm{c}}<0$ and ${\lambda }_{\mathrm{c},0}<{\lambda }_{\mathrm{c}}$ such that, for each $\lambda \in ({\lambda }_{\mathrm{c},0},{\lambda }_{\mathrm{c}})\cup ({\lambda }_{\mathrm{c}},\infty )$, $H(\lambda )$ is diagonalized by a proper Bogoliubov transformation, so that the spectrum of $H(\lambda )$ is explicitly identified, where the spectrum of $H(\lambda )$ for $\lambda >{\lambda }_{\mathrm{c}}$ is different from that for $\lambda \in ({\lambda }_{\mathrm{c},0},{\lambda }_{\mathrm{c}})$. As for the case $\lambda <{\lambda }_{\mathrm{c},0}$, we show that $H(\lambda )$ is unbounded from above and below. In the massless case, ${\lambda }_{\mathrm{c}}$ coincides with ${\lambda }_{\mathrm{c},0}$.

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