Orthogonal Polynomials of Several Variables and Symmetry Groups

经典的一元正交多项式涉及到数学的多个方面，例如：切比雪夫、勒让德和厄米特多项式分别用于近似理论、高斯求积和氢原子量子模型。这些多项式之所以如此有效，源于相关的微分方程式、递推关系和权函数。这个项目的目标是构造和研究类似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种非常有效的方法。该方案考虑的主要类型是单形、超立方体和二十面体的对称群。前两个由维度标记的无穷族组成，并且与允许任意坐标排列或符号变化的系统相关联。二十面体是一种三维规则固体，具有真正古老的历史记录，这是目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的形状允许无限数量的分子刚性地装配在一起，即使二十面体不允许这种图案(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角形对称)。这个项目涉及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分析；‘调和分析’一词的起源是将音符分解为基音和泛音的技术。这项技术已经发展到诸如将量子力学系统分解成不同能级，或者用容易计算的多项式来逼近函数等应用。除了二十面体群，该方案还涉及超八面体群的球谐多项式的模拟，它既允许坐标的排列也允许坐标的符号变化。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以及自旋Calogero模型(一维空间中的相同粒子，每个粒子的自旋为+/-1)中的应用。它的目标是找到足够具体的公式，以允许渐近计算：对系统长期行为的定性描述。最后，计划研究例外微分方程，因为一些基本参数具有导致异常现象的特定孤立值；一方面是Korteweg-DeVries方案中的双谱方程，另一方面是“超可积”量子力学系统。该方案的技术基础是由作者构造的与反射群相关的微分-差分算子，并已在许多物理和数学研究文章中得到应用。

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.