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

TL;DR

We address the revival problem for the one-dimensional periodic Schrödinger equation with piecewise $C^2$ potential. By combining Sturm–Liouville spectral theory, variational bounds, and Prüfer-phase asymptotics, we derive precise eigenvalue and eigenfunction behavior under piecewise smooth potentials. We prove a continuous revival formula at rational times $t=2\pi q/r$, $u(t,x)= w(t,x) + \psi(t,x)$, where $\psi$ is a finite sum of translations of the initial data and $w$ is a continuous remainder; the revival component is described explicitly in terms of $a_n,b_n$, the Fourier basis associated with the problem. Numerical simulations corroborate the theoretical predictions for piecewise-constant potentials, demonstrating the robustness of dispersive quantization and extending revival phenomena to non-smooth settings.

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.

Figures (3)

• Figure 1: The real part of the solution.
• Figure 2: The real part of the revival component and the remainder.
• Figure 3: The imaginary part of the solution, the revival component, and the remainder.

Theorems & Definitions (30)

• Definition 1.1: Periodic Schrödinger equation
• Theorem 1.1: Continuous revival at rational times
• Lemma 2.1
• proof
• Theorem 2.1
• proof
• Theorem 2.2: Sturm Comparison Theorem
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5