1994

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

## Publication

### Publication

*
J. Math. Phys.
,
Volume 35
p. 113-
126
*

The members *H*_{w}=**p**^{2} +*V*_{w}(**x**)_{ }of ergodic families of random operators, defined in *h*=*L*^{2}(R* ^{d}*), are considered as acting in fibers in a direct integral decomposition of

*k*=

*L*

^{2}(W,

*P*(

*d*w);

*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

*L*

^{2}(W,

*P*)

**.**

Additional Metadata | |
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J. Math. Phys. | |

Tip, A. (1994). Random time evolution and direct integrals: constants of the motion and the mass operator. J. Math. Phys., 35, 113–126. |