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In this paper we continue the study of the energy-momentum spectrum of aclass of translation invariant, linearly coupled, and massive Hamiltonians fromnon-relativistic quantum field theory. The class contains the Hamiltonians ofE. Nelson and H. Froehlich. One of us previously investigated the structure ofthe ground state mass shell and the bottom of the continuous energy-momentumspectrum. Here we study the continuous energy-momentum spectrum itself up tothe two-boson threshold, the threshold for energetic support of two-bosonscattering states. We prove that non-threshold embedded mass shells have finitemultiplicity and can accumulate only at thresholds. We furthermore establishthe non-existence of singular continuous energy-momentum spectrum. Our resultshold true for all values of the particle-field coupling strength but only belowthe two-boson threshold. The proof revolves around the construction of acertain relative velocity vector field used to construct a conjugate operatorin the sense of Mourre.

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