Continuous Revival of the Periodic Schrödinger Equation with Piecewise $C^2$ Potentials

Abstract

In this paper, we investigate the revivals of the one-dimensional periodic Schrödinger equation with a piecewise $C^2$ potential function. As has been observed through numerical simulations of the equation with various initial data and potential functions, the solution, while remaining fractalized at irrational times, exhibits a form of revival at rational times. The goal is to prove that the solution at these rational times is given by a finite linear combination of translations and dilations of the initial datum, plus an additional continuous term, which we call "continuous revival". In pursuit of this result, we present a review of relevant properties of the periodic Schrödinger equation as an eigenvalue problem, including asymptotic results on both the eigenvalues and eigenfunctions.

Revised arguments in Section 3.2, main results unchanged. 26 pages, 24 figures. Comments are welcome!