小时间可控薛定谔方程的例子
Examples of small-time controllable Schrödinger equations
AI总结：
本文首次给出小时间近似可控的双线性薛定谔方程例子，通过控制二次势频率实现空间膨胀，进而产生时间收缩，将大时间控制结果推广到小时间。
摘要：
已知多种物理相关的双线性薛定谔方程在大时间下是近似可控的。然而，存在一些在大时间下近似可控但在小时间下不可控的例子。这种障碍例如出现在存在（次）二次势的情况下，因为高斯态至少在小时间内被保持。在这项工作中，我们提供了小时间近似可控的双线性薛定谔方程的首批例子。特别地，我们表明对二次势频率的控制允许构造沿空间膨胀任意快速演化的近似解。一旦我们能够进行空间膨胀，我们就可以利用它们来产生时间收缩。通过这种方式，我们基于先前的大时间控制结果，获得了小时间控制。
英文摘要：
A variety of physically relevant bilinear Schrödinger equations are known to be approximately controllable in large times. There are however examples which are approximately controllable in large times, but not in small times. This obstruction happens e.g. in the presence of (sub)quadratic potentials, because Gaussian states are preserved, at least for small times. In this work, we provide the first examples of small-time approximately controllable bilinear Schrödinger equations. In particular, we show that a control on the frequency of a quadratic potential permits to construct approximate solutions that evolve arbitrarily fast along space-dilations. Once we have access to space-dilations, we can exploit them to generate time-contractions. In this way, we build on previous results of large-time control, to obtain control in small times.
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