具有周期势的薛定谔方程的规范不变冻结高斯近似方法
Gauge-invariant frozen Gaussian approximation method for the Schrödinger equation with periodic potentials
AI总结：
本文提出了一种规范不变的冻结高斯近似方法，用于求解具有周期势的线性薛定谔方程，在半经典极限下，该方法基于相空间的渐近分析和布洛赫波，有效捕捉解的高频振荡，且在光程和束扩散情况下均有效，并避免了计算规范依赖的伯尔利相位的困难。
摘要：
我们开发了一种规范不变的冻结高斯近似（GIFGA）方法，用于求解具有周期势的线性薛定谔方程（LSE）在半经典极限下的问题。该方法将Herman-Kluk传播器推广到具有周期介质的情况。它基于相空间的渐近分析和布洛赫波，提供了一种高效的计算工具，以捕捉解的高频振荡。与几何光学和高斯束方法相比，GIFGA在光程和束扩散两种情况下均有效。此外，它对布洛赫本征函数的规范选择具有不变性，因此避免了计算规范依赖的伯尔利相位的数值困难。我们通过多个一维例子对方法进行了数值测试，特别是验证了阶数为一的收敛性，这与我们的配套分析论文[Delgadillo, Lu and Yang, arXiv:1504.08051]一致。
英文摘要：
We develop a gauge-invariant frozen Gaussian approximation (GIFGA) method for the linear Schrödinger equation (LSE) with periodic potentials in the semiclassical regime. The method generalizes the Herman-Kluk propagator for LSE to the case with periodic media. It provides an efficient computational tool based on asymptotic analysis on phase space and Bloch waves to capture the high-frequency oscillations of the solution. Compared to geometric optics and Gaussian beam methods, GIFGA works in both scenarios of caustics and beam spreading. Moreover, it is invariant with respect to the gauge choice of the Bloch eigenfunctions, and thus avoids the numerical difficulty of computing gauge-dependent Berry phase. We numerically test the method by several one-dimensional examples, in particular, the first order convergence is validated, which agrees with our companion analysis paper [Delgadillo, Lu and Yang, arXiv:1504.08051].
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