S-transformation is a unitary transformation accomplishing the main part of the diagonalization of the interaction Hamiltonian in strong coupling theories. For a compact symmetry group the action of the S-transformation on the integrals of motion is derived.

Among approaches to defining the Feynman integral, various sequential methods are probably the most popular ones. Here we consider that one based on product formulae, the original idea of which belongs to Nelson71'. This method is powerful in the sense that it enables to prove the strict version of Feynman's basic dynamical formula for a wide class of poten tials^^/including time-dependent ones/4/in comparison with other known methods (see, eg, refs./5, 6/). 0n the other, hand, the method does not correspond exactly to Feynman's heuristic prescription because it replaces the exact action along poly gonal paths by its Riemannian approximation. Moreover, one can regard as an unsatisfactory feature that it employs the" equidistant" time-interval partitions only. In the present paper we show that the last mentioned sour ce of arbitrariness can be removed: the product formula for perturbations of propagators by

The Feynman maps introduced first by Truman are examined. The domain considered here consists of the Fresnel-integrable functions in the sense of Albeverio and Hoegh-Krohn; extensions to wider classes of functions will be studied in the following paper. The original definition of the F-maps is slightly modified: we start from the underlying measures on the Hilbert space of paths in order to avoid use of improper integrals. Some new properties of the F-maps are derived. In particular, the dominated convergence theorem is shown to be not valid for the F₁-map (or Feynman integral); this fact is of a certain importance for the classical limit of quantum mechanics.

The present paper which represents the fourth part of the series devoted to analysis of a simple Lee-type model of two-particle decay deals with three problems. The first one concerns relation of the model to the scattering theory. We prove asymptotic completeness for the elastic scattering of the two light particles and show that for a sufficiently weak coupling this system has just one resonance whose position is the same as that of the pole which yield the main contribution to the decay law. The second problem concerns spectral concentration; we prove its occurrence for families of intervals around<i>E</i> that shrink slower than quadratically in<i>g</i>. Finally, necessary and sufficient conditions for the existence of bound states are discussed.

We consider a simple model which is a caricature of a crystal interacting with a radiation field. The model has two bands of continuous spectrum and the particle can pass from the upper one to the lower by radiating a photon, the coupling between the excited and deexcited states being of a Friedrichs type. Under suitable regularity and analyticity assumptions we find the continued resolvent and show that for weak enough coupling it has a curve-type singularity in the lower halfplane which is a deformation of the upper-band spectral cut. We then find a formula for the decay amplitude and show that for a fixed energy it is approximately exponential at intermediate times, while the tail has a power-like behaviour.