Ballistic transport for limit-periodic Schrödinger operators in one dimension

In this paper, we consider the transport properties of the class of limit-periodic continuum Schr\"odinger operators whose potentials are approximated exponentially quickly by a sequence of periodic functions. For such an operator \(H\), and \(X_H(t)\) the Heisenberg evolution of the position operator, we show the limit of \(\frac{1}{t}X_H(t)\psi\) as \(t\to\infty\) exists and is nonzero for \(\psi\ne 0\) 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.