Revivals and fractalisation in the linear free space Schrödinger equation
Revivals and fractalisation in the linear free space Schrödinger equation
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线性自由空间薛定谔方程中的复兴和分形
DOI：
10.1090/qam/1547
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发表时间：
2018
影响因子：
0.8
通讯作者：
David A. Smith
中科院分区：
文献类型：
--
作者：
P. Olver;N. Sheils;David A. Smith
本文考虑一维线性自由空间薛定谔方程在齐次线性边界条件下的解。我们证明，在伪周期边界条件的情况下，初边值问题的解决方案表现出在特定的（“理性”）时间的复苏现象，这意味着它是一个线性组合的初始数据的一定数量的副本。等价地，基本解在这些时候是δ函数的有限线性组合。在其他（“不合理”）时间，对于适当粗略的初始数据，例如，阶跃或更一般的分段常数函数，该解表现出连续但类似分形的轮廓。此外，我们表示的解决方案一般齐次线性边界条件的数值计算的特征函数。使用均匀变换方法（UTM）推导出替代解决方案公式，可以证明在更一般的情况下是有用的。然后，我们调查的一般线性边界条件，包括罗宾的影响，并找到新的“耗散”复苏的情况下，能量下降的条件。
We consider the one-dimensional linear free space Schrodinger 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.