Graph and Group Classification of Terms in Adiabatic and Semiclassical Expansions

Abstract

The author describe three ways in which concepts from discrete mathematics are being used, along with computer algebra systems, to bring order into the welter of terms that appear when asymptotic expansions of propagators or heat kernels are carried out to high orders. (The key needs are (a) specifying a linearly independent set of terms, and (b) classifying terms by their structural properties.) (1) For a Schrodinger operator with a scalar potential, a closed-form solution has been given as a sum over graphs (including loops and multiple lines). To generate the series requires only the listing of all graphs with given numbers of lines and vertices, taking account of isomorphism. This can be done by Polya-type methods, obtaining along the way a finer classification of graphs by the distribution of lines among the vertices. (2) For problems involving a Riemann curvature tensor, enumerating and classifying all the linearly independent terms of each order involves the Schur-Young-Littlewood theory of representations of the symmetric and orthogonal groups. (3) For scalar equations, the connectedness of graphs has a profound relation to the mathematical structure of the series. If there is an analogue of this principle for equations with matrix-valued potentials, the relevant notion of connectedness must involve commutators as well as index contractions.