A functional-analytic approach to the steady, one-speed neutron transport equation with anisotropic scattering
A functional-analytic approach to the steady, one-speed neutron transport equation with anisotropic scattering
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具有各向异性散射的稳定单速中子输运方程的泛函分析方法
DOI：
10.1002/cpa.3160270404
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发表时间：
1974
影响因子：
3
通讯作者：
E. Larsen
中科院分区：
文献类型：
--
作者：
E. Larsen
奇异本征函数方法自1960年由Case首次应用以来，得到了广泛的关注[1]。在他的原始论文中，Case考虑了具有各向同性散射的时间无关、单速度、一维输运方程。从那时起，其他作者将凯斯的技术扩展到更一般的问题。特别是米卡[!]我发展了一种研究各向异性散射的方法。后来，McCormick和Kuscer[3]，[4]通过用伴随特征函数表示各向异性半程公式来改进Mika的结果。然而，关于奇异特征函数技术仍然存在许多问题，包括各向异性伴随特征函数的存在性（参见[4]）。此外，将奇异特征函数技术置于严格的基础上所做的工作很少。据笔者所知，仅有Larsen b[5]、Larsen and Habetler b[6]和Hangelbroek b[7]发表了这样的著作。这三篇论文处理的是最初由Case考虑的简化输运方程。所有处理这个问题的方法都是将传输算子分离为空间导数和一个算子，我们称之为KI，它只作用于角变量（见式（2.4））。在前两篇文章中，直接从K的解算符推导出K的谱性质，后一篇文章采用C*-代数方法求解谱问题。在本文中，我们推广了前几篇论文的方法来解释可表示为有限勒让德多项式和的各向异性散射核。除了统一地推导出特征函数和全、半量程公式外，我们的方法还得到了一些新的结果。例如，满值域和半值域公式成立的函数类被证明包含在u= 0处奇异的函数（见定理1和定理2）。讨论了这一点的意义。导出了伴随特征函数，并证明了伴随特征函数在c< 1时的存在性。众所周知
The singular eigenfunction approach to transport theory has received wide attention since its first application by Case in 1960 [l]. In his original paper, Case considered the time independent, one speed, one-dimensional transport equation with isotropic scattering. Since then other authors have extended Case’s technique to more general problems. In particular, Mika [! I developed an approach to anisotropic scattering. Later, McCormick and Kuscer [3],[4] refined Mika’s results by expressing the anisotropic half range formulas in terms of adjoint eigenfunctions. However many questions remain regarding the singular eigenfunction technique, including the very existence of the anisotropic adjoint eigenfunctions (cf.[4]). In addition, very little work has been done to place the singular eigenfunction technique on a rigorous basis. To this author’s knowledge the only such work has been by Larsen [5], Larsen and Habetler [6], and Hangelbroek [7]. These three papers deal with the simplified transport equation originally considered by Case. All treat the problem by separating the transport operator into the space derivative and an operator, which we call KI, acting only on the angular variable (see equation (2.4)). In the former papers, the spectral properties of K are deduced directly from the resolvent operator of K. The latter paper adopts a C*-algebra approach to the spectral problem. In this paper, we generalize the approach of the former papers to account for an anisotropic scattering kernel expressible as a finite sum of Legendre polynomials. Aside from deriving the eigenfunctions and the full and half range formulas in a unified way, our approach yields a number of new results. For instance, the class of functions for which the full and half range formulas hold is shown to include functions which are singular at, u= 0 (see Theorems 1 and 2). The meaning of this is discussed. The adjoint eigenfunctions are derived and their existence is proved for c< 1. As is known