线性与非线性薛定谔方程中受扰谐振子的傅里叶方法
Fourier methods for the perturbed harmonic oscillator in linear and nonlinear Schrödinger equations
AI总结：
针对含时谐振势或微扰的Gross-Pitaevskii方程，提出一种结合傅里叶方法与精确求解含时谐振子的新数值方法，在保持高效性的同时提高精度。
摘要：
我们考虑具有由含时谐振势或其小扰动给出的势阱的Gross-Pitaevskii方程的数值积分。分裂方法常与傅里叶技术结合使用，因为系统可以分裂为动能部分和剩余部分，且每部分可以使用快速傅里叶变换高效求解。将系统分裂为量子谐振子问题和剩余部分，在许多情况下可以获得更高的精度，但这需要在Hermite基函数和坐标空间之间切换，并且对于含时频率或强非线性而言效率不高。我们展示了如何构建新方法，这些方法结合了傅里叶方法的优点，同时精确求解含时谐振子（或通过使用Magnus积分器和适当的分解实现高精度）。
英文摘要：
We consider the numerical integration of the Gross-Pitaevskii equation with a potential trap given by a time-dependent harmonic potential or a small perturbation thereof. Splitting methods are frequently used with Fourier techniques since the system can be split into the kinetic and remaining part, and each part can be solved efficiently using Fast Fourier Transforms. To split the system into the quantum harmonic oscillator problem and the remaining part allows to get higher accuracies in many cases, but it requires to change between Hermite basis functions and the coordinate space, and this is not efficient for time-dependent frequencies or strong nonlinearities. We show how to build new methods which combine the advantages of using Fourier methods while solving the timedependent harmonic oscillator exactly (or with a high accuracy by using a Magnus integrator and an appropriate decomposition).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
