Computational Search Methods for Non-trivial Zeros

Author: Robert Tuzun (SUNY Brockport, Department of Computational Science)

Abstract

Wigner 3n-j symbols are commonly used in scattering theory, particle physics, spectroscopy, and many other areas of chemistry and physics. Zero values are indicative of symmetry relations, and as such point the way both to fast, accurate computational schemes and to algebraic structure within the quantum theory of angular momentum. Although symmetry relations discovered by Regge in 1958 explain most of these zeros, other instances (known as non-trivial zeros) remain unexplained. A computational search of 3-j symbols for J at or below 1800 (6652619045416 3-j symbols unrelated by Regge symmetry) yielded 595564 non-trivial zeros. This search was undertaken by relating the expression for 3-j symbols to alternating binomial sums, then by using recursion among the binomial sums to find zero values. Patterns in the non-trivial zeros, and in "false" zeros culled from lists of likely candidates, suggest possible strategies for pursuing faster recursion schemes than are currently available.