Geometric spectral analysis

在原子和分子物理学中，基本方程已经建立。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。这些复杂的问题本质上是数学的或计算的。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。进一步的简化是可能的，当潜在的研究是一个家庭的先前研究的潜力的包络曲线。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”。对于相对论非常重要的中等能量的少粒子系统，如原子和分子的困难，导致了光谱等价的非相对论模型的构建。这可以通过从数据生成模型的迭代反演方法来实现。因此，这项工作的一个重要组成部分是计算技术的发展，无论是精确的计算机代数，并通过直接数值分析。例如，在渐近迭代法（AIM）中，计算机程序迭代函数序列，直到达到精确或近似解。由于数学的普遍性，这种方法对于比它最初创建的科学问题更广泛的科学问题是有用的。

In atomic and molecular physics, the fundamental equations have already been established. What it is often difficult is to say unambiguously what the implications of the theory are for specific physical systems. These problems of complexity 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. Further simplifications are possible when 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. The difficulties for few-particle systems such as atoms and molecules at medium energies, where relativity is important, lead to the construction of spectrally equivalent non-relativistic models. This may be effected by an iterative inversion method that generates the model from the data. Thus an important component of this work is the development of computational techniques, both with exact computer algebra, and by direct numerical analysis. For example, in the Asymptotic Iteration Method (AIM), a computer program iterates a sequence of functions until an exact or approximate solution is reached. Because of the universality of mathematics, this method is useful for a much wider class of scientific problems than that for which it was originally created.