Ballistic transport for limit-periodic Schr\"odinger operators in one dimension

TL;DR

This paper proves strong ballistic transport in one-dimensional limit-periodic Schrödinger operators, showing that the position operator's Heisenberg evolution exhibits nonzero linear growth for a dense set of initial states, indicating quadratic growth of the second moment.

Contribution

It establishes the first proof of ballistic transport in continuum almost periodic non-periodic Schrödinger operators with exponentially approximated potentials.

Findings

Limit of the scaled position operator exists and is nonzero.

Second moment of the position grows quadratically in time.

Ballistic transport is proven for a dense set of initial states.

Abstract

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.