On Darboux's Approach to $R$-Separability of Variables

We discuss the problem of $R$-separability (separability of variables with a factor $R$) in the stationary Schrödinger equation on $n$-dimensional Riemann space. We follow the approach of Gaston Darboux who was the first to give the first general treatment of $R$-separability in PDE (Laplace equation on $\mathbb E^3$). According to Darboux $R$-separability amounts to two conditions: metric is isothermic (all its parametric surfaces are isothermic in the sense of both classical differential geometry and modern theory of solitons) and moreover when an isothermic metric is given their Lamé coefficients satisfy a single constraint which is either functional (when $R$ is harmonic) or differential (in the opposite case). These two conditions are generalized to $n$-dimensional case. In particular we define $n$-dimensional isothermic metrics and distinguish an important subclass of isothermic metrics which we call binary metrics. The approach is illustrated by two standard examples and two less standard examples. In all cases the approach offers alternative and much simplified proofs or derivations. We formulate a systematic procedure to isolate $R$-separable metrics. This procedure is implemented in the case of 3-dimensional Laplace equation. Finally we discuss the class of Dupin-cyclidic metrics which are non-regularly $R$-separable in the Laplace equation on $\mathbb E^3$.