 08130 C. G rard, A. Panati
 Spectral and scattering theory for \\ spacecutoff
$P(\varphi)_{2}$ models with variable metric
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Jun 26, 08

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Abstract. We consider spacecutoff $P(\varphi)_{2}$ 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.
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