# Revivals and Fractalisation in the Linear Free Space Schr\"odinger Equation

## TL;DR

This paper studies the linear free space Schr"odinger equation on a bounded interval, revealing revival phenomena at rational times and fractal profiles at irrational times, with new insights into boundary conditions and solution representations.

## Contribution

It demonstrates revival phenomena under pseudoperiodic boundary conditions, introduces new solution formulas using the Uniform Transform Method, and explores effects of general linear boundary conditions including Robin.

## Key findings

- Revival at rational times as linear combinations of initial data
- Fractal-like solution profiles at irrational times for rough initial data
- Novel dissipative revivals under energy-decreasing boundary conditions

## Abstract

We consider the one-dimensional linear free space Schr\"odinger equation on a bounded interval subject to homogeneous linear boundary conditions. We prove that, in the case of pseudoperiodic boundary conditions, the solution of the initial-boundary value problem exhibits the phenomenon of revival at specific (`rational') times, meaning that it is a linear combination of a certain number of copies of the initial datum. Equivalently, the fundamental solution at these times is a finite linear combination of delta functions. At other (`irrational') times, for suitably rough initial data, e.g., a step or more general piecewise constant function, the solution exhibits a continuous but fractal-like profile. Further, we express the solution for general homogenous linear boundary conditions in terms of numerically computable eigenfunctions. Alternative solution formulas are derived using the Uniform Transform Method (UTM), that can prove useful in more general situations. We then investigate the effects of general linear boundary conditions, including Robin, and find novel `dissipative' revivals in the case of energy decreasing conditions.