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Article Dans Une Revue Annales Henri Poincaré Année : 2008

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

Résumé

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.
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Dates et versions

hal-00290676 , version 1 (26-06-2008)

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Christian Gérard, Annalisa Panati. Spectral and scattering theory for space-cutoff $P(\varphi)_{2}$ models with variable metric. Annales Henri Poincaré, 2008, 9 (8), pp.1575-1629. ⟨10.1007/s00023-008-0396-2⟩. ⟨hal-00290676⟩
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