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

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.

References in corpus (5)

• Asymptotic Completeness in Quantum Field Theory: Translation Invariant Nelson Type Models Restricted to the Vacuum and One-Particle Sectors
• Second order perturbation theory for embedded eigenvalues
• Lower Spectral Branches of a Particle Coupled to a Bose Field
• Regularity of Eigenstates in Regular Mourre Theory
• Absence of Embedded Mass Shells: Cerenkov Radiation and Quantum Friction

Cited by in corpus (7)

• On the domain of the Nelson Hamiltonian
• The translation invariant massive Nelson model: III. Asymptotic completeness below the two-boson threshold
• Effective mass of the polaron -- revisited
• A Taylor-like Expansion of a Commutator with a Function of Self-Adjoint, Pairwise Commuting Operators
• Spectral analysis of the spin-boson Hamiltonian with two bosons for arbitrary coupling and bounded dispersion relation
• Mourre theory and spectral analysis of energy-momentum operators in relativistic quantum field theory
• Differential Equations of Quantum Mechanics