基于奇异近似的直线上散射
Scattering on the line via singular approximation
摘要：
出于应用程序的声学成像，目前的工作建立了一个框架来分析散射的一维波，亥姆霍兹，薛定谔和Riccati方程，允许系数比以前的理论可以容纳的更奇异。代替标准散射矩阵或Weyl-Titchmarsh $m$ -函数，分析集中在一个新的对象，广义反射系数，映射频率（或频谱参数）的庞加莱磁盘的自同构。广义反射系数的奇异形式，这是服从于直接分析，服务于近似的一般情况。单位圆和单位圆盘上的正交多项式起着关键的技术作用，PSL $(2,\mathbb{R})$上的奇异黎曼结构也是如此。新定义的调和指数算子也起着核心作用，它被引入来调解阻抗（或折射率）和反射系数之间的关系。该方法导致新的，明确的公式和有效的算法，正向和反向散射。该算法可以被看作是FFT的非线性模拟。此外，散射关系示出为基本的在一个精确的意义上在或低于连续阻抗的临界阈值。然而，对于不连续阻抗，反射系数在无穷远处停止衰减，经典的迹公式被打破，并且散射关系由于概周期结构的出现而变得复杂。
英文摘要：
Motivated by applications to acoustic imaging, the present work establishes a framework to analyze scattering for the one-dimensional wave, Helmholtz, Schrödinger and Riccati equations that allows for coefficients which are more singular than can be accommodated by previous theory. In place of the standard scattering matrix or the Weyl-Titchmarsh $m$-function, the analysis centres on a new object, the generalized reflection coefficient, which maps frequency (or the spectral parameter) to automorphisms of the Poincaré disk. Purely singular versions of the generalized reflection coefficient, which are amenable to direct analysis, serve to approximate the general case. Orthogonal polynomials on both the unit circle and unit disk play a key technical role, as does an exotic Riemannian structure on PSL$(2,\mathbb{R})$. A central role is also played by the newly-defined harmonic exponential operator, introduced to mediate between impedance (or index of refraction) and the reflection coefficient. The approach leads to new, explicit formulas and effective algorithms for both forward and inverse scattering. The algorithms may be viewed as nonlinear analogues of the FFT. In addition, the scattering relation is shown to be elementary in a precise sense at or below the critical threshold of continuous impedance. For discontinuous impedance, however, the reflection coefficient ceases to decay at infinity, the classical trace formula breaks down, and the scattering relation is complicated by the emergence of almost periodic structure.
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