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

Abstract

The members Hω=p2+Vω(x) of ergodic families of random operators, defined in ℋ=L2(Rd), are considered as acting in fibers in a direct integral decomposition of 𝒦=L2(Ω,P(dω);ℋ), (Ω,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 Σ(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 Ĥ, which can be decomposed, Ĥ→Ĥ(k), in a way that diagonalizes p. Thus problems about averaged time evolution can be formulated entirely in terms of Ĥ(k) in L2(Ω,P).