Orthogonal Polynomials of Several Variables and Symmetry Groups

经典的单变量正交多项式在数学的许多方面都有重要的意义，例如：Chebyshev多项式、Legendre多项式和Hermite多项式分别用于近似理论、高斯正交和氢原子量子模型中。这些多项式如此有效的原因来自于相关的微分方程、递归关系和权函数。这个项目的目标是构造和研究同样有用的多变量正交多项式族。研究具有有限对称群的系统是实现这一目标的一个非常有效的方法。在这个建议中考虑的主要类型是单纯形、超立方体和二十面体的对称群。前两个包含无限族，由维度标记，并且与允许坐标任意排列或符号变化的系统相关联。二十面体是一种具有真正古老历史记录的三维规则固体，是当前准晶体研究的主题。这些都是矛盾的结构；像盐这样的普通晶体具有一种允许无限数量的分子刚性组装在一起的形状，即使二十面体不允许这种模式（称为规则镶嵌），也存在具有二十面体结构的物理物质（例如，在电子显微镜观察下出现五边形对称）。这个项目是关于寻找多项式的正交基；这些基具有特殊的意义，因为它们可以直接用于函数的分析；“和声分析”一词的起源是将一个音符分解成基本音和泛音的技术。这项技术已经发展到将量子力学系统分解为不同的能级，或通过易于计算的多项式近似函数等应用中。除了二十面体群外，本文还讨论了球谐多项式对高八面体群的模拟，它允许坐标的置换和符号变化。其中一个目标是研究正交多项式在具有三体相互作用的量子系统中的应用，以及自旋Calogero模型（一维空间中的相同粒子，每个自旋为+/- 1）。目标是找到足够具体的公式来允许渐近计算：对系统长期行为的定性描述。最后，计划研究异常微分方程，在某种意义上，一些潜在的参数采取特定的孤立值导致异常现象；一方面存在Korteweg-deVries格式中的双谱方程，另一方面存在“超可积”量子力学系统。该方法的技术基础是与反射群相关的微分-差分算子，该算子已在许多物理和数学研究中得到应用。

The classical orthogonal polynomials of one variable are significantly involved in several aspects of mathematics, for example: the Chebyshev, Legendre, and Hermite polynomials are used in approximation theory, Gaussian quadrature, and the hydrogen atom quantum model, respectively. The reasons why these polynomials are so effective come from the associated differential equations, the recurrence relations, and the weight functions. The goal of this project is to construct and study similarly useful families of orthogonal polynomials of several variables. A very fruitful approach to this aim is the study of systems which have finite symmetry groups. The main types considered in this proposal are the symmetry groups of the simplex, the hypercube and the icosahedron. The first two comprise infinite families, labeled by dimension, and are associated with systems allowing arbitrary permutations of the coordinates, or sign-changes as well. The icosahedron is a three-dimensional regular solid with a truly ancient historical record, which is the subject of current research in quasi-crystals. These are paradoxical structures; ordinary crystals like salt have a shape which allows the rigid fitting together of an unlimited number of molecules, and even though the icosahedron does not allow such patterns (called regular tessellations) there exist physical substances with icosahedral structures (for example, pentagonal symmetry appears under electron microscope observations).This project is concerned with finding orthogonal bases of polynomials; these bases have a special significance because they can be used directly in the analysis of functions; the origin of the phrase ``harmonic analysis'' is the technique of decomposing a musical note into fundamentals and overtones. This technique has developed into such applications as decompositions of quantum-mechanical systems into different energy levels, or the approximations of functions by easily computable polynomials. Besides the icosahedral group, the proposal deals with the analog of spherical harmonic polynomials for the hyperoctahedral group, which allows both permutations as well as sign-changes of the co-ordinates. One of the aims is to investigate the application of orthogonal polynomials to quantum systems with three-body interactions as well as the spin Calogero models (identical particles in a one-dimensional space, each with a spin of +/- 1). It is a goal to find formulas specific enough to allow asymptotic calculations: qualitative descriptions of the long-term behavior of the systems. Finally, it is planned to investigate exceptional differential equations, in the sense that some underlying parameters take on specific isolated values leading to unusual phenomena; on the one hand there are the bi-spectral equations in the Korteweg-deVries scheme, and on the other hand there are ``super-integrable'' quantum-mechanical systems. The technical foundation of this proposal is the differential-difference operator associated with reflection groups which was constructed by the proposer and has been used in many research articles in physics and mathematics.