Yuri Kozitsky
Irreducible Dynamics of Quantum Systems Associated with L{\'e}vy Processes
(39K, AMS-LaTeX)

ABSTRACT.  Quantum systems described by the Schr\"odinger operators $H = 
\sum_{j=1}^N \mathit{\Phi}(p_j) + W(x_1 , \dots, x_N)$, $p_j = 
-\imath \mathit{\nabla}_j$, $x_j \in \mathbb{R}^\nu$ with 
$\mathit{\Phi}$ being continuous functions such that the 
pseudo-differential operators $\mathit{\Phi}(p_j)$ generate L{\'e}vy 
processes, are considered. It is proven that the linear span of the 
operators $\alpha_{t_1}(F_1) \cdots \alpha_{t_n} (F_n)$ is dense in 
the algebra of all observables in the $\sigma$-strong and hence in 
the $\sigma$-weak and strong topologies. Here $\alpha_t (F) = 
\exp(\imath tH) F \exp(- \imath tH)$ are time automorphisms and 
$F$'s are taken from families of multiplication operators obeying 
conditions described in the paper. This result implies that a linear 
functional continuous in either of these topologies is fully 
determined by its values on such products. In the case of KMS states 
this yields a representation of such states in terms of path 
integrals.