Geometric spectral analysis

几何光谱分析

• 批准号: 3438-2008
• 负责人: Hall, Richard
• 金额: $ 1.55万
• 依托单位: Concordia University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2012
• 资助国家: 加拿大
• 起止时间: 2012-01-01 至 2013-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=497968
• 关键词: Geometric; spectral; analysis

In many fields of low-energy physics, such as atomic and molecular physics, the fundamental equations are established, or at least agreed upon. What it is often difficult is to say unambiguously what the implications of the theory are for specific physical systems. The problems are essentially mathematical or computational. This research is principally to do with developing methods that can be used to reduce complexity and allow the problems to be solved to a satisfactory degree of accuracy. Systems of identical particles in quantum mechanics obey a very powerful constraint: under the interchange of particles, the allowed states are either symmetric (bosons) or anti-symmetric (fermions). A formulation can be devised which is a dynamic analog of the crystallographic principle, that the crystal reproduces the shape of the molecular unit cell; dynamically, a many-particle quantum system behaves approximately like a special scaled version of a two-particle system. Thus we can build many-body effects into the parameters of a representative model that is simple enough to be solved analytically. Further simplifications are possible if the interaction potential is a smooth transformation of a potential for which exact solutions are already known. This leads to representations in which the potential studied is an envelope curve of a family of previously-studied potentials. Thus symmetry and geometry are used to effect computational simplifications, which ideally retain definite relationships between the original problem and the soluble model. The validity of these methods is often independent of the number of particles or the dimension of the space in which they move. Fundamental questions naturally arise to do with the relation of one problem to another, so that new 'comparison theorems' are called for. It is also an essential part of this work to develop computational techniques, both with exact computer algebra, and direct numerical analysis. In one method under development, called the Asymptotic Iteration Method (AIM), a computer program iterates a set of functions until an exact or approximate solution is reached.

在低能物理的许多领域，例如原子和分子物理，基本方程是建立的，或者至少是达成一致的。通常很难毫不含糊地说出这一理论对特定物理系统的含义。这些问题本质上是数学或计算问题。这项研究主要是为了开发可以用来降低复杂性并使问题解决到令人满意的精度程度的方法。量子力学中的完全相同的粒子系统服从一个非常强大的约束：在粒子互换的情况下，允许的态要么是对称的(玻色子)，要么是反对称的(费米子)。可以设计出一种公式，它是结晶学原理的动态模拟，即晶体再现分子单胞的形状；动态地，多粒子量子系统的行为近似于两粒子系统的特殊比例版本。因此，我们可以将多体效应构建到一个具有代表性的模型的参数中，该模型足够简单，可以用解析的方法求解。如果相互作用势是已知精确解的势的光滑变换，则进一步简化是可能的。这导致了所研究的势是先前研究的势族的包络曲线的表示。因此，对称性和几何被用来实现计算简化，这理想地保留了原始问题和可解模型之间的明确关系。这些方法的有效性通常与粒子的数量或它们在其中运动的空间的维度无关。与一个问题与另一个问题的关系有关的基本问题自然而然地出现了，因此需要新的“比较定理”。开发计算技术也是这项工作的重要组成部分，既有精确的计算机代数，也有直接的数值分析。在一种正在开发中的方法中，称为渐近迭代法(AIM)，计算机程序迭代一组函数，直到得到精确或近似的解。