The properties of mean values of random variable with values in the set of semigroups of unitary operators are investigated. The mean value of random semigroup has no semigroup property. But it is equivalent in Chernoff sense to the semigroup with generator which is the result of averaging of generators of values of random semigroup. The problem of ambiguity of quantization of Hamiltonian systems is studied by using of semigroup averaging procedure.In particular the wide class of Shchrodinger operators on the graph is described by using of semigropus averaging procedure. The equivalence of presentation of quantum dynamics by unitary semigroup and by pseudomeasure on the space of trajectories of classical system is proved. The properties of averaging of random measures and random pseudomeasures (in particular, the linear and non-linear functionals on the space of measures and pseudomeasures) are studied.

Quantization, Chernoff theorem, Chernoff equivalence, random semigroups

Mathematical modelling in actual problems of science and technics