Spectral and scattering theory for space-cutoff $P(\varphi)_{2}$ models with variable metric

We consider space-cutoff $P(\varphi)_{2}$ models with a variable metric of the form $$H= \d\G(\omega)+ \int_{\rr}g(x):P(x, \varphi(x)):\d x,$$ on the bosonic Fock space $L^{2}(\rr)$, where the kinetic energy $\omega= h^{\12}$ is the square root of a real second order differential operator $$h= Da(x)D+ c(x),$$ where the coefficients $a(x), c(x)$ tend respectively to 1 and $m_{\infty}^{2}$ at $\infty$ for some $m_{\infty}>0$. The interaction term $\int_{\rr}g(x):P(x, \varphi(x)):\d x$ is defined using a bounded below polynomial in $\lambda$ with variable coefficients $P(x, \lambda)$ and a positive function $g$ decaying fast enough at infinity. We extend in this paper the results of \cite{DG} where $h$ had constant coefficients and $P(x, \lambda)$ was independent of $x$. We describe the essential spectrum of $H$, prove a Mourre estimate outside a set of thresholds and prove the existence of asymptotic fields. Our main result is the {\em asymptotic completeness} of the scattering theory, which means that the CCR representation given by the asymptotic fields is of Fock type, with the asymptotic vacua equal to bound states of $H$. As a consequence $H$ is unitarily equivalent to a collection of second quantized Hamiltonians. An important role in the proofs is played by the {\em higher order estimates}, which allow to control powers of the number operator by powers of the resolvent. To obtain these estimates some conditions on the eigenfunctions and generalized eigenfunctions of $h$ are necessary. We also discuss similar models in higher space dimensions where the interaction has an ultraviolet cutoff.