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}+\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)≥0 a space-cutoff and P(λ,λ) 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