Exact wave functions for concentric two-electron systems

We show that the exact solution of the Schr\"odinger equation for two electrons confined to two distinct concentric rings or spheres can be found in closed form for particular sets of the ring or sphere radii. In the case of two concentric rings, we report exact polynomial and irrational solutions. The same methodology is applied to the case of two concentric spheres for which we report exact polynomial solutions for the ground state and the excited states of $S$ symmetry. For these concentric systems, we show that the exact wave function does not contain terms proportional to the interelectronic distance due to the spatial separation of the electrons.