C. G rard, A. Panati
Spectral and scattering theory for \\ space-cutoff 
$P(\varphi)_{2}$ models with variable metric
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ABSTRACT.  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.