正弦函数带余弦姿态
Sine function with a cosine attitude
摘要：
我们给出了一个揭示性的揭露，解决了一个重要的问题，在散射理论中，如何构建两个渐近正弦解的波动方程的相移使用相同的基础具有相同的边界条件在原点。这些解决方案的解析级数表示。在1D中，其中一个解是一个偶函数，其行为渐近为sin（x），而另一个解是一个奇函数，其行为渐近为cos（x）。后者在原点处为零，而前者的导数在原点处为零。消除级数的最低N项使得这些函数在原点周围的一个区间内变小，该区间的大小随N而增加。我们采用的J-矩阵散射方法的工具，在这些解决方案的建设在一个和三个维度。
英文摘要：
We give a revealing expose that addresses an important issue in scattering theory of how to construct two asymptotically sinusoidal solutions of the wave equation with a phase shift using the same basis having the same boundary conditions at the origin. Analytic series representations of these solutions are obtained. In 1D, one of the solutions is an even function that behaves asymptotically as sin(x), whereas the other is an odd function, which is asymptotically cos(x). The latter vanishes at the origin whereas the derivative of the former becomes zero there. Eliminating the lowest N terms of the series makes these functions vanishingly small in an interval around the origin whose size increases with N. We employ the tools of the J-matrix method of scattering in the construction of these solutions in one and three dimensions.
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