The translation Invariant Massive Nelson Model: II. The Continuous Spectrum Below the Two-boson Threshold

TL;DR

This paper analyzes the continuous energy-momentum spectrum of translation invariant quantum field Hamiltonians below the two-boson threshold, proving finite multiplicity of embedded mass shells and absence of singular continuous spectrum.

Contribution

It extends previous work by characterizing the structure of the spectrum up to the two-boson threshold, including multiplicity and spectral type results.

Findings

Non-threshold embedded mass shells have finite multiplicity.

Embedded mass shells can only accumulate at thresholds.

The continuous spectrum has no singular continuous component.

Abstract

In this paper we continue the study of the energy-momentum spectrum of a class of translation invariant, linearly coupled, and massive Hamiltonians from non-relativistic quantum field theory. The class contains the Hamiltonians of E. Nelson and H. Froehlich. One of us previously investigated the structure of the ground state mass shell and the bottom of the continuous energy-momentum spectrum. Here we study the continuous energy-momentum spectrum itself up to the two-boson threshold, the threshold for energetic support of two-boson scattering states. We prove that non-threshold embedded mass shells have finite multiplicity and can accumulate only at thresholds. We furthermore establish the non-existence of singular continuous energy-momentum spectrum. Our results hold true for all values of the particle-field coupling strength but only below the two-boson threshold. The proof revolves around the construction of a certain relative velocity vector field used to construct a conjugate operator in the sense of Mourre.