Random time evolution and direct integrals: constants of the motion and the mass operator

The members H_w=p² +V_w(x) of ergodic families of random operators, defined in h=L²(R^d), are considered as acting in fibers in a direct integral decomposition of k=L²(W, P(dw);h), (W,P) being the underlying probability space. Such a formulation may turn out to be useful in providing a rigorous background for theoretical physical methods employed in treating random systems. As an example a rigorous definition of the mass operator S(z) is presented and the formulation of transport equations in a functional analytic setting is briefly indicated. In case there is translational invariance in the probability law the operator H is shown to be unitarily equivalent to an operator o(^,H), which can be decomposed, o(^,H) Æ o(^,H)(k), in a way that diagonalizes p. Thus problems about averaged time evolution can be formulated entirely in terms of o(^,H) (k) in L²(W, P).