2018/12/20 by Peter J. Olver, Olver, Peter J, Natalie E. Sheils +3 · 2 citations
Physics and Astronomy · Mathematics · Engineering · #Model Reduction and Neural Networks #Fractional Differential Equations Solutions #Electromagnetic Simulation and Numerical Methods
paper · pdf · doi:10.48550/arxiv.1812.08637
We consider the one-dimensional linear free space Schr "odinger equation on a\nbounded interval subject to homogeneous linear boundary conditions. We prove\nthat, in the case of pseudoperiodic boundary conditions, the solution of the\ninitial-boundary value problem exhibits the phenomenon of revival at specific\n(`rational') times, meaning that it is a linear combination of a certain number\nof copies of the initial datum. Equivalently, the fundamental solution at these\ntimes is a finite linear combination of delta functions. At other\n(`irrational') times, for suitably rough initial data, e.g., a step or more\ngeneral piecewise constant function, the solution exhibits a continuous but\nfractal-like profile. Further, we express the solution for general homogenous\nlinear boundary conditions in terms of numerically computable eigenfunctions.\nAlternative solution formulas are derived using the Uniform Transform Method\n(UTM), that can prove useful in more general situations. We then investigate\nthe effects of general linear boundary conditions, including Robin, and find\nnovel `dissipative' revivals in the case of energy decreasing conditions.\n