Geometric spectral analysis

在低能物理的许多领域，如原子和分子物理，基本方程已经建立，或者至少已经达成一致。通常困难的是明确地说出该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性的。这项研究主要是与开发方法有关，这些方法可以用来降低复杂性，并允许以令人满意的精度解决问题。在量子力学中，相同粒子的系统服从一个非常强大的约束：在粒子交换下，允许的状态要么是对称的（玻色子），要么是反对称的（费米子）。可以设计一种公式，它是晶体学原理的动态模拟，即晶体再现分子单位细胞的形状；动态地说，一个多粒子量子系统的行为近似于一个特殊的双粒子系统的缩放版本。因此，我们可以将多体效应建立到一个简单到可以解析求解的代表性模型的参数中。如果相互作用势是已知精确解的势的平滑变换，则进一步的简化是可能的。这导致所研究的势是先前研究过的势族的包络曲线的表示。因此，使用对称和几何来实现计算简化，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运动的空间的维度无关。关于一个问题与另一个问题的关系，自然会产生一些基本问题，因此需要新的“比较定理”。发展计算技术也是这项工作的重要组成部分，包括精确的计算机代数和直接的数值分析。在一种正在开发的方法中，称为渐近迭代法（AIM），计算机程序迭代一组函数，直到达到精确或近似解。

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.