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arXiv · 2504.09397

Continuous Revival of the Periodic Schr\"{o}dinger Equation with Piecewise $C^2$ Potentials

Dinh-Quan Tran·Peter J. Olver
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Abstract

In this paper, we investigate the revivals of the one-dimensional periodic Schr\"odinger 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 Schr\"odinger equation as an eigenvalue problem, including asymptotic results on both the eigenvalues and eigenfunctions.

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Dinh-Quan Tran, Peter J. Olver. 2025-04-13. Continuous Revival of the Periodic Schr\"{o}dinger Equation with Piecewise $C^2$ Potentials. https://arxiv.org/abs/2504.09397

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