Separation of variables in the Hamilton–Jacobi, Schrödinger, and related equations. I. Complete separation

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Abstract
It was established by Levi‐Civita that in n dimensions there exist n+1 types of coordinate systems in which the Hamilton–Jacobi equation is separable, n of which are in general nonorthogonal; the form of the separated equations was given by Burgatti and Dall’Acqua. In this paper first the general forms of the n+1 types of metric tensors of the corresponding corresponding Riemannian spaces Vn are determined. Then, sufficient conditions are given for coordinate systems in which the Schrödinger, Helmholtz, and Laplace equation are separable. It is shown that there again exist n+1 types of such systems, whose metric tensors are of the same form as those of the Hamilton–Jacobi equation. However, except for the ’’essentially geodesic case’’ of Levi‐Civita they are further restricted by a condition on the determinant of the metric; this condition is a generalization of that found by Robertson for orthogonal systems in the case of the Schrödinger equation.