Abstract
We study counting processes introduced by Davies [11] on general state spaces. The concept of a refinement of a counting process (CP), corresponding to the possibility of distinguishing particles, for instance according to their energy or phase, is introduced, and refinements of general CP's are classified. Then CP's with bounded interaction rate are classified on general state spaces, and sufficient conditions are given in order that the operators characterizing the interaction rate can be formulated in the Schrödinger picture. For CP's with unbounded interaction rates it is shown that analogous to the case of bounded interaction rates there is a family of operators characterizing the interaction rate. Commutation relations for such processes are derived. For constructions of CP's with unbounded interaction rate it is shown that it essentially suffices to solve the semigroup perturbation problem. Finally refinements of these CP's are characterized by “measures”E→J(E) on the set of different particles, where eachJ(E) in an (unbounded) operator.
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Müller, B. On generalized quantum stochastic counting processes. Commun.Math. Phys. 173, 453–474 (1995). https://doi.org/10.1007/BF02101654
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DOI: https://doi.org/10.1007/BF02101654