Averaging of quantum dynamical semigroups

In the framework of the elliptic regularization method, the Cauchy problem for the Schrödinger equation with discontinuous degenerating coefficients is associated with a sequence of regularized Cauchy problems and the corresponding regularized dynamical semigroups. We study a divergent sequence of quantum dynamical semigroups as a random process with values in the space of quantum states defined on a measurable space of regularization parameters with a finitely additive measure. The mathematical expectation of the considered processes determined by the Pettis integral defines a family of averaged dynamical transformations. We investigate the semigroup property and the injectivity and surjectivity of the averaged transformations. We establish the possibility of defining the process by its mathematical expectation at two different instants and propose a procedure for approximating an unknown initial state by solutions of a finite set of variational problems on compact sets.