Degani Ilan
Mathematics Institute, University of Bergen, Johannes Brunsgate 12, Bergen 5008, Norway.
J Chem Phys. 2008 Apr 28;128(16):164108. doi: 10.1063/1.2899018.
The efficiency of a numerical method can be greatly improved by combining it with coordinate transformations tailored to a given problem. This is the basis for the mapped Fourier methods. However, obtaining "good" coordinate transformations is a major obstacle for this approach in multidimensions. Here, we calculate coordinate transformations based on solving the Monge-Ampere equation. These transformations are combined in the mapped Fourier method and applied to Schrodinger's equation in multidimensions. Dramatic improvements in accuracy compared to the standard Fourier method were observed in eigenvalue calculations for two-dimensional systems. This work indicates that the Monge-Ampere equation may serve as a useful tool for constructing efficient representations for problems in computational quantum mechanics and other fields.
通过将数值方法与针对特定问题定制的坐标变换相结合，其效率可得到极大提高。这是映射傅里叶方法的基础。然而，在多维情况下，获得“良好”的坐标变换是这种方法的主要障碍。在此，我们基于求解蒙日 - 安培方程来计算坐标变换。这些变换被应用于映射傅里叶方法，并用于求解多维薛定谔方程。在二维系统的本征值计算中，与标准傅里叶方法相比，精度有了显著提高。这项工作表明，蒙日 - 安培方程可能是为计算量子力学及其他领域的问题构建有效表示的有用工具。
