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