Schroedinger equation on a generic radial grid

In this note, we discuss the choice of radial grid in the numerical resolution of the Schroedinger equation. We detail the transformation of the equation resulting from a change of variable and function for a generic radial grid, using either the explicit or implicit form of the relation describing the change of variable, and apply it to the $ar+b\ln (r)$ log-linear mesh. It is shown that, in the former case, the first three derivatives of the Lambert $W$ function are required. This complication becomes unnecessary if we adopt the implicit relation instead.