Spectral and scattering theory of charged $P(\varphi)_2$ models

We consider in this paper space-cutoff charged $P(\varphi)_{2}$ models arising from the quantization of the non-linear charged Klein-Gordon equation: $$(\p_{t}+\i V(x))^{2}\phi(t, x)+ (-\Delta_{x}+ m^{2})\phi(t,x)+ g(x)\p_{\overline{z}}P(\phi(t,x), \overline{\phi}(t,x))=0,$$ where $V(x)$ is an electrostatic potential, $g(x)\geq 0$ a space-cutoff and $P(\lambda, \overline{\lambda})$ a real bounded below polynomial. We discuss various ways to quantize this equation, starting from different CCR representations. After describing the construction of the interacting Hamiltonian $H$ we study its spectral and scattering theory. We describe the essential spectrum of $H$, prove the existence of asymptotic fields and of wave operators, and finally prove the {\em asymptotic completeness} of wave operators. These results are similar to the case when V=0.