The phenomenon of revivals on complex potential Schr\"odinger's equation

TL;DR

This paper rigorously characterizes revivals in complex potential Schrödinger equations, showing explicit solutions at rational times as combinations of initial data transformations, clarifying a historically observed but poorly defined phenomenon.

Contribution

It provides a rigorous definition and explicit description of revivals in Schrödinger equations with complex potentials, expanding understanding of this phenomenon.

Findings

Revivals occur at rational times as finite combinations of translations and dilations.

Solutions include an explicit discrete part plus a continuous term.

The results apply under a regularity condition in a broad class of Schrödinger equations.

Abstract

The mysterious phenomena of revivals in linear dispersive periodic equations was discovered first experimentally in optics in the 19th century, then rediscovered several times by theoretical and experimental investigations. While the term has been used systematically and consistently by many authors, there is no consensus on a rigorous definition. In this paper, we describe revivals modulo a regularity condition in a large class of Schr\"odinger's equations with complex bounded potentials. As we show, at rational times the solution is given explicitly by finite linear combinations of translations and dilations of the initial datum, plus an additional continuous term.