It is stated in many books that analytic closed solutions to the time-independent electronic Schrödinger equation, $$\hat{H}\Psi = E\Psi, $$ exist for the one-electron problem (e.g. hydrogen atom, assuming separability of nuclear and electronic motion) but that such solutions do not exist for systems with more than one-electron and thus approximation methods are required to solve the equation.
Specifically, on going from a one-electron system to a two-electron system, with fixed nuclei, something changes that makes closed analytic solution of the equation no longer possible.
Clearly this is related to the inter-electronic interaction because closed analytic solutions are possible for systems of non-interacting particles. Many resources suggest that the many-electron problem is "too difficult" to solve analytically but do not give any further details. This raises the question: is it the case that closed analytic solutions cannot exist, or that they could exist, but it is very difficult to find them? And, if they cannot exist, then how is this determined?
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