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We study the existence and the continuity properties of the boundary values on the real axis of the resolvent of a self-adjoint operator H in the framework of the conjugate operator method initiated by Mourre. We allow the conjugate operator A to be the generator of a C 0 -semigroup (finer estimates require A to be maximal symmetric) and we consider situations where the first commutator ½H; iA is not comparable to H: The applications include the spectral theory of zero mass quantum field models. r 2004 Elsevier Inc. All rights reserved.

In recent years, the spectral properties of the translation invariant Nelson model has been studied. Some of the results obtained did not extend to the related polaron model for technical reasons related to the typical assumption of boundedness of the phonon dispersion relation in the polaron model. In this paper, we work with a large class of linearly coupled translation invariant models which includes both the Nelson model and H. Fröhlich's polaron model. The problems considered are chosen based on relevance for the polaron model. A key input is an analysis of the behavior of the bottom of the spectrum of the fiber Hamiltonians at large total momentum.

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We discuss Hilbert space-valued stochastic differential equations associated with the heat semi-groups of the standard model of non-relativistic quantum electrodynamics and of corresponding fiber Hamiltonians for translation invariant systems. In particular, we prove the existence of a stochastic flow satisfying the strong Markov property and the Feller property. To this end we employ an explicit solution ansatz. In the matrix-valued case, i.e., if the electron spin is taken into account, it is given by a series of operator-valued time-ordered integrals, whose integrands are factorized into annihilation, preservation, creation, and scalar parts. The Feynman-Kac formula implied by these results is new in the matrix-valued case. Furthermore, we discuss stochastic differential equations and Feynman-Kac representations for an operator-valued integral kernel of the semi-group. As a byproduct we obtain analogous results for Nelson's model.