Geometric spectral analysis

在许多低能物理学领域，如原子和分子物理学，基本方程是建立起来的，或者至少是一致同意的。 通常困难的是，要明确地说出这个理论对特定物理系统的意义。这些问题本质上是数学或计算问题。 这项研究主要是开发方法，可以用来降低复杂性，并允许问题得到解决，以一个令人满意的精度。 量子力学中的全同粒子系统服从一个非常强大的约束：在粒子交换下，允许的态要么是对称的（玻色子），要么是反对称的（费米子）。 可以设计一个公式，它是晶体学原理的动态模拟，即晶体再现分子晶胞的形状;动态地，多粒子量子系统的行为近似于双粒子系统的特殊缩放版本。 因此，我们可以将多体效应构建到一个代表性模型的参数中，这个模型足够简单，可以用解析法求解。 如果相互作用势是已知精确解的势的光滑变换，则进一步的简化是可能的。 这导致代表性的潜在研究是一个家庭的包络曲线以前研究的潜力。 因此，对称性和几何形状被用来影响计算简化，这在理想情况下保留了原始问题和可解模型之间的确定关系。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。 基本问题自然会出现，与一个问题与另一个问题的关系有关，因此需要新的“比较定理”。 开发计算技术（包括精确计算机代数和直接数值分析）也是这项工作的重要组成部分。 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.