Averaging of random semigroups and the ambiguity of quantization of Hamiltonian systems

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 semigroups averaging procedure.