On the essential spectrum of the translation invariant Nelson model

which implies a UV-cutoff. Here O(ν) denotes the orthogonal group. The physically interesting choices for the dispersion relations Ω and ω are Ω(η) = η/2M , Ω(η) = √ η2 + M2 and ω(k) = √ k2 + m2, where M,m > 0 are the electron and boson masses. We will however work with general forms of both Ω and ω. As for ω, this is partly motivated by the similarity with the Polaron model, cf. [5, 11], where ω is not explicitly known. We make no attempt here to say anything about the Polaron model. The operator H commutes with the total momentum p⊗ 1lF + 1lK⊗ dΓ (k) and hence fibers as H ∼ Rν H(ξ)dξ, where the fiber Hamiltonians H(ξ), ξ ∈ R , are operators on F given by