Toeplitz波和Hankel波的无反射性质：用Bessel函数分析
The reflectionless properties of Toeplitz waves and Hankel waves: an analysis via Bessel functions
摘要：
我们研究在一个空间维度和时间的波动方程的边界处的无反射性质，在一个众所周知的矩阵，从一个简单的空间离散化。已知的是，在该设置中表示拉普拉斯算子的熟悉的第二差分矩阵的所有矩阵函数是Toeplitz矩阵和Hankel矩阵的和。波动方程的解就是这样一个矩阵函数。在这里，我们研究相应的波，我们称之为Toeplitz波和汉克尔波的行为。我们表明，这些波可以写为某些线性组合的第一类甚至贝塞尔函数。我们发现这些波的精确和明确的公式。我们还表明，Toeplitz和汉克尔波是无反射的，分别为奇数，遍历域。我们的分析自然提出了一种新的计算机模拟方法，允许控制，因此可以提前选择反射的数量。从我们的分析中得出的一个有吸引力的结果是著名的移位矩阵的出现，以及其他可能被认为是移位矩阵的汉克尔版本的矩阵。通过揭示的代数结构的解决方案中的移位矩阵，我们清楚地表明，Toeplitz和汉克尔波确实是无反射的边界上的偶数或奇数遍历。虽然无反射边界条件的问题有很长的历史，但我们认为，我们在这里采用的矩阵函数的观点是新的。
英文摘要：
We study reflectionless properties at the boundary for the wave equation in one space dimension and time, in terms of a well-known matrix that arises from a simple discretisation of space. It is known that all matrix functions of the familiar second difference matrix representing the Laplacian in this setting are the sum of a Toeplitz matrix and a Hankel matrix. The solution to the wave equation is one such matrix function. Here, we study the behaviour of the corresponding waves that we call Toeplitz waves and Hankel waves. We show that these waves can be written as certain linear combinations of even Bessel functions of the first kind. We find exact and explicit formulae for these waves. We also show that the Toeplitz and Hankel waves are reflectionless on even, respectively odd, traversals of the domain. Our analysis naturally suggests a new method of computer simulation that allows control, so that it is possible to choose -- in advance -- the number of reflections. An attractive result that comes out of our analysis is the appearance of the well-known shift matrix, and also other matrices that might be thought of as Hankel versions of the shift matrix. By revealing the algebraic structure of the solution in terms of shift matrices, we make it clear how the Toeplitz and Hankel waves are indeed reflectionless at the boundary on even or odd traversals. Although the subject of the reflectionless boundary condition has a long history, we believe the point of view that we adopt here in terms of matrix functions is new.
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