TY - JOUR

T1 - REVIVALS AND FRACTALISATION IN THE LINEAR FREE SPACE SCHRÖDINGER EQUATION

AU - OLVER, PETER J.

AU - SHEILS, NATALIE E.

AU - SMITH, DAVID A.

N1 - Publisher Copyright:
©2019 Brown University

PY - 2020/6

Y1 - 2020/6

N2 - . We consider the one-dimensional linear free space Schrödinger 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 Unified 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.

AB - . We consider the one-dimensional linear free space Schrödinger 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 Unified 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.

UR - http://www.scopus.com/inward/record.url?scp=85087152662&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85087152662&partnerID=8YFLogxK

U2 - 10.1090/QAM/1547

DO - 10.1090/QAM/1547

M3 - Article

AN - SCOPUS:85087152662

SN - 0033-569X

VL - 78

SP - 161

EP - 192

JO - Quarterly of Applied Mathematics

JF - Quarterly of Applied Mathematics

IS - 2

ER -