Geometric spectral analysis

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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.