Ballistic transport for limit-periodic Schrodinger operators in one dimension

In this paper, we consider the transport properties of the class of limit-periodic continuum Schrodinger operators whose potentials are approximated exponentially quickly by a sequence of periodic functions. For such an operator H, and XH (t) the Heisenberg evolution of the position operator, we show the limit of t1 XH(t) as t ->infinity exists and is nonzero for psi not equal 0 belonging to a dense subspace of initial states which are sufficiently regular and of suitably rapid decay. This is viewed as a particularly strong form of ballistic transport, and this is the first time it has been proven in a continuum almost periodic non-periodic setting. In particular, this statement implies that for the initial states considered, the second moment grows quadratically in time.