Random operators and quantum mechanics

Random operator is studied as the measurable map of a probability space into a set of operators endowed with some topology and corresponding Borel sigma-algebra. The notion of random operator is applied to the investigation of initial boundary value problems. The approximation of singular initial boundary value problem defines a random variable with values in the set of operators or operator semigroups. In this framework the model of degenerate Hamiltonian and its regularization are studied. We investigate the dynamics on some $C*$-subalgebras of the algebra of bounded linear operators generated by a degenerate Hamiltonian. Conservation or destruction of the semigroup property for the mean value of random semigroup is investigated. The procedure of reconstruction of the semigroup property by means of the Feynman-Chernoff iteration is constructed. Conditions of validity and violation of the law of large numbers for compositions of independent identically distributed random semigroups are obtained.