Robertson Condition

For the Helmholtz Differential Equation to be Separable in a coordinate system, the Scale Factors $h_i$ in the Laplacian

$$\nabla^2=\sum_{i=1}^3 {1\over h_1h_2h_3} {\partial\over\par... ...ft({{h_1h_2h_3\over{h_i}^2}{\partial\over\partial u_i}}\right)$$ (1)

and the functions $f_i(u_i)$ and $\Phi_{ij}$ defined by

$${1\over f_n}{\partial\over\partial u_n}\left({f_n{\partial X... ...t) +({k_1}^2\Phi_{n1}+{k_2}^2\Phi_{n2}+{k_3}^2\Phi_{n3})X_n=0$$ (2)

must be of the form of a Stäckel Determinant

$$S=\vert\Phi_{mn}\vert = \left\vert\matrix{ \Phi_{11} & \Phi_... ...{33}\cr}\right\vert={h_1h_2h_3\over f_1(u_1)f_2(u_2)f_3(u_3)}.$$ (3)

See also Helmholtz Differential Equation, Laplace's Equation, Separation of Variables, Stäckel Determinant

References

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part 1. New York: McGraw-Hill, p. 510, 1953.