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

We consider in this paper space-cutoff charged $P(φ)_{2}$ models arising from the quantization of the non-linear charged Klein-Gordon equation: \[ (\p_{t}+ıV(x))^{2}ϕ(t, x)+ (-Δ_{x}+ m^{2})ϕ(t,x)+ g(x)\p_{\overline{z}}P(ϕ(t,x), \overlineϕ(t,x))=0, \] where $V(x)$ is an electrostatic potential, $g(x)\geq 0$ a space-cutoff and $P(λ, \overlineλ)$ 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.