Separation of variables in the Hamilton–Jacobi, Schrödinger, and related equations. II. Partial separation

Abstract

Sufficient conditions are given for coordinate systems in which the Hamilton–Jacobi equation and the Schrödinger and related equations are partially separable in n dimensions. For the first equation, the solution is assumed to be a sum of ν+τ functions of a single variable, and of μ[0⩽μ⩽ (n−ν−τ)/2] groups of other variables; for the other equations, products of such functions are assumed. These assumptions lead to ν+τ completely separated differential equations, of which ν are linear in the separation constants, and μ partially separated equations depending on the remaining n−ν−τ variables. The general forms of the various metric tensors g^kl of the Riemannian spaces V_n as well as of the allowed potentials V corresponding to the different possible types of such equations are determined; they are identical for the Hamilton–Jacobi equation and for the other equations studied, except that for the latter some of the metrics are further restricted by a condition on their determinants. The results are established by methods similar to those used in Paper I of this series for complete separation, and include the results obtained there as special cases. In the course of determining the allowed forms of the g^kl it is also established that there exist ν+τ+μ independent first integrals linear and quadratic in the momenta for the dynamical systems descrived by the Hamilton–Jacobi or the Schrödinger equation. The ν linear ones are homogeneous, and the τ+μ quadratic ones correspond to homogeneous quadratic integrals of the geodesics of the V_n. These results imply the existence of ν Killing vectors and of τ+μ Killing tensors of rank two for the V_n. Further polynomial integrals can be constructed; those integrals of degree r which are independent of the original ν+τ+μ integrals each correspond to an independent Killing tensor of rank r.