Spectral and scattering theory of charged $P(\varphi)_2$ models

TL;DR

This paper develops spectral and scattering theory for charged $P()_2$ quantum field models, establishing essential spectrum, asymptotic fields, wave operators, and asymptotic completeness.

Contribution

It introduces new quantization methods for charged $P()_2$ models and proves spectral and scattering properties, including asymptotic completeness.

Findings

Describes the essential spectrum of the Hamiltonian.

Proves existence of asymptotic fields and wave operators.

Establishes asymptotic completeness of wave operators.

Abstract

We consider in this paper space-cutoff charged $P(\varphi)_{2}$ models arising from the quantization of the non-linear charged Klein-Gordon equation: \[ (\p_{t}+\i V(x))^{2}\phi(t, x)+ (-\Delta_{x}+ m^{2})\phi(t,x)+ g(x)\p_{\overline{z}}P(\phi(t,x), \overline{\phi}(t,x))=0, \] where $V(x)$ is an electrostatic potential, $g(x)\geq 0$ a space-cutoff and $P(\lambda, \overline{\lambda})$ a real bounded below polynomial. We discuss various ways to quantize this equation, starting from different CCR representations. After describing the construction of the interacting Hamiltonian $H$ we study its spectral and scattering theory. We describe the essential spectrum of $H$, prove the existence of asymptotic fields and of wave operators, and finally prove the {\em asymptotic completeness} of wave operators. These results are similar to the case when V=0.