插值尺度函数在波包传播中的应用
An application of interpolating scaling functions to wave packet propagation
AI总结：
研究基于插值尺度函数（ISF）的波包传播方法，该方法具有对角势能矩阵和稀疏动能矩阵，并通过简单代数变换精确计算哈密顿矩阵元，数值模拟表明其准确高效。
摘要：
研究了在插值尺度函数（ISF）基组中的波包传播。ISF在基于样条双正交小波的多分辨率分析中广为人知。ISF构成对应于等距网格的基数基组。它们具有由用于生成ISF的底层插值多项式确定的紧支撑大小。在该基组中，势能矩阵是对角的，动能矩阵是稀疏的，并且在1D情况下具有带状对角结构。该基组的一个重要特征是哈密顿量的矩阵元可以通过简单代数变换精确计算，且数值实现高效。因此，网格点数和底层插值多项式的阶数可以轻松变化，从而允许以规则方式接近伪谱方法的精度，类似于高阶有限差分方法。H+H2共线碰撞的数值模拟结果表明，ISF为波包传播方法提供了准确且高效的表示。
英文摘要：
Wave packet propagation in the basis of interpolating scaling functions (ISF) is studied. The ISF are well known in the multiresolution analysis based on spline biorthogonal wavelets. The ISF form a cardinal basis set corresponding to an equidistantly spaced grid. They have compact support of the size determined by the underlying interpolating polynomial that is used to generate ISF. In this basis the potential energy matrix is diagonal and the kinetic energy matrix is sparse and, in the 1D case, has a band-diagonal structure. An important feature of the basis is that matrix elements of a Hamiltonian are exactly computed by means of simple algebraic transformations efficiently implemented numerically. Therefore the number of grid points and the order of the underlying interpolating polynomial can easily be varied allowing one to approach the accuracy of pseudospectral methods in a regular manner, similar to high order finite difference methods. The results of numerical simulations of an H+H_2 collinear collision show that the ISF provide one with an accurate and efficient representation for use in the wave packet propagation method.
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