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

TL;DR

This paper extends spectral and scattering theory results for space-cutoff $P()_{2}$ models with variable metric, proving asymptotic completeness and describing the essential spectrum, under more general conditions than previous work.

Contribution

It generalizes previous results by allowing variable coefficients in the kinetic operator and polynomial interactions, establishing asymptotic completeness and spectral properties.

Findings

Describes the essential spectrum of the Hamiltonian.

Proves Mourre estimate outside thresholds.

Establishes asymptotic completeness of the scattering theory.

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.