Spherical harmonic matching with a derivative jump (${\Phi }_n^{\mathrm{in}}=a_n{r}^n{P}_n,{\Phi }_n^{\mathrm{out}}=a_n{r}^{-n-1}{P}_n$)

For a regular interior and a decaying exterior harmonic potential with continuous values on the unit sphere, the same coefficient multiplies the two radial branches of each Legendre polynomial mode. Its exterior-minus-interior radial derivative is $-(2n+1)a_n{P}_n$. Thus an axisymmetric prescribed jump $j_n{P}_n(\cos \theta )$ produces coefficient $a_n=-j_n/(2n+1)$. This diagonal matching rule reduces a spherical interface problem to its angular-mode coefficients.