We study the Schr\"odinger equation $i\partial_t\psi=-\Delta\psi+V\psi$ on $L^2((0,1),\mathbb{C})$ where $V$ is a very high and localized potential wall. We aim to perform permutations of the eigenmodes and to control the solution of the equation. We consider the process where the position and the height of the potential wall change as follows. First, the potential increases from zero to a very large value, so a narrow potential wall is formed that almost splits the interval into two parts; then the wall moves to a different position, after which the height of the wall decays to zero again. We show that even though the rate of the variation of the potential's parameters can be arbitrarily slow, this process alternates adiabatic and non-adiabatic dynamics, leading to a non-trivial permutation of the eigenstates. Furthermore, we consider potentials with several narrow walls and we show how an arbitrarily slow motion of the walls can lead the system from any given state to an arbitrarily small neighborhood of any other state, thus proving the approximate controllability of the above Schr\"odinger equation by means of a soft, quasi-adiabatic variation of the potential.

中文翻译：

---

通过势壁的准绝热运动置换量子本征模式

我们在 $L^2((0,1),\mathbb{C})$ 上研究 Schr\"odinger 方程 $i\partial_t\psi=-\Delta\psi+V\psi$ 其中 $V$ 是非常高和局部的势壁。我们的目标是执行本征模式的置换并控制方程的解。我们考虑势壁的位置和高度变化的过程如下。首先，势从零增加到一个非常大的值，所以形成了一个狭窄的势墙，几乎将间隔分成两部分；然后墙移动到不同的位置，之后墙的高度再次衰减为零。我们表明，即使势参数的变化可以任意缓慢，这个过程交替绝热和非绝热动力学，导致本征态的非平凡排列。此外，我们考虑具有几个窄壁的势能，并展示了壁的任意慢速运动如何将系统从任何给定状态引导到任何其他状态的任意小邻域，从而证明上述 Schr\"odinger 方程的近似可控性为势的软，准绝热变化的手段。