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A level set method for the semiclassical limit of the Schr?dinger equation with discontinuous potentials

机译：具有不连续电势的薛定inger方程半经典极限的水平集方法

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摘要

We propose a level set method for the semiclassical limit of the Schr?dinger equation with discontinuous potentials. The discontinuities in the potential corresponds to potential barriers, at which incoming waves can be partially transmitted and reflected. Previously such a problem was handled by Jin and Wen using the Liouville equation - which arises as the semiclassical limit of the Schr?dinger equation - with an interface condition to account for partial transmissions and reflections (S. Jin, X. Wen, SIAM J. Num. Anal. 44 (2006) 1801-1828). However, the initial data are Dirac-delta functions which are difficult to approximate numerically with a high accuracy. In this paper, we extend the level set method introduced in (S. Jin, H. Liu, S. Osher, R. Tsai, J. Comp. Phys. 210 (2005) 497-518) for this problem. Instead of directly discretizing the Delta functions, our proposed method decomposes the initial data into finite sums of smooth functions that remain smooth in finite time along the phase flow, and hence can be solved much more easily using conventional high order discretization schemes.Two ideas are introduced here: (1) The solutions of the problems involving partial transmissions and partial reflections are decomposed into a finite sum of solutions solving problems involving only complete transmissions and those involving only complete reflections. For problems involving only complete transmission or complete reflection, the method of JLOT applies and is used in our simulations; (2) A reinitialization technique is introduced so that waves coming from multiple transmissions and reflections can be combined seamlessly as new initial value problems. This is implemented by rewriting the sum of several delta functions as one delta function with a suitable weight, which can be easily implemented numerically. We carry out numerical experiments in both one and two space dimensions to verify this new algorithm.
机译：我们为具有不连续电位的薛定？方程的半经典极限提出了一种水平集方法。电位的不连续性对应于势垒，在该势垒处，入射波可以被部分地透射和反射。以前，Jin和Wen使用Liouville方程（作为Schr？dinger方程的半经典极限出现）解决了这个问题，并采用了界面条件来解释部分透射和反射（S. Jin，X。Wen，SIAM J分子分析44（2006）1801-1828）。然而，初始数据是狄拉克-德尔塔（Dirac-delta）函数，其难以以高精度在数值上近似。在本文中，我们针对此问题扩展了（S. Jin，H. Liu，S. Osher，R. Tsai，J.Comp。Phys。210（2005）497-518）中介绍的水平集方法。我们提出的方法不是直接离散化Delta函数，而是将初始数据分解为平滑函数的有限总和，这些有限函数沿着相流在有限时间内保持平滑，因此可以使用常规的高阶离散化方案轻松解决。在此介绍：（1）将涉及部分透射和部分反射的问题的解决方案分解为解决仅涉及完全透射和仅涉及完全反射的问题的有限总和。对于仅涉及完全透射或完全反射的问题，JLOT方法适用于我们的仿真; （2）引入了一种重新初始化技术，使来自多次透射和反射的波可以无缝组合为新的初始值问题。这是通过将多个增量函数的和重写为具有适当权重的一个增量函数来实现的，可以很容易地通过数字实现。我们在一个和两个空间维度上进行了数值实验，以验证该新算法。

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