Knowledge Bank

“Asymptotic methods for the treatment of asymmetric rotor problems in the domain of high J and low $K1$ have been generalized in two ways: (1) A non-linear differential equation with periodic coefficients takes the place of the linear Mathieu type equation, which has first been used by $Golden2$ to calculate approximately asymmetric rotor eigenvalues, and which has served as starting point to derive asymptotic formulas in reference 1. The characteristic matrix of this non-linear equation is infinite, but its elements are exactly identical with the elements of a reduced form of the asymmetric rotor matrix within their domain of definition. (2) A procedure permitting extension of asymptotic expansions for eigenvalues and eigenfunctions of the Mathieu equation into the domain of noticeable symmetry type splitting has recently been developed by Dingle and $Mueller.3$ This procedure is now adapted to the derivation of the corresponding asymptotic formulas in asymmetric rotor theory for energy levels and transitions for which symmetry-type splittings are noticeable but still small. The inclusion of such symmetry type corrections extends appreciably the domain of applicability of the asymptotic methods.”