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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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    $\begingroup$ A lesson from recent history: arxiv.org/abs/1203.5408 $\endgroup$
    – Count Iblis
    Commented Jul 4, 2014 at 20:56
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    $\begingroup$ I've come across a book called "Beautiful models 70 years of exactly solved quantum many-body problems". Probably you can find some clue there. $\endgroup$
    – Pu Zhang
    Commented Jul 4, 2014 at 21:02
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    $\begingroup$ Really important fact: "closed-form" and "analytic" as used colloquially are basically meaningless. What you mean is "subjectively simple expression in terms of functions that I like." Is the sine function closed-form? It is if you say it is, it isn't if you say it isn't. After all, sine is only computable in the limit of infinitely many arithmetic operations, but we gave it a special name. So perfectly well-defined solutions to well-posed differential equations can be non-"analytic" to some simply because no one bothered to give those solutions a special name. $\endgroup$
    – user10851
    Commented Jul 4, 2014 at 23:08

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A more precise mathematical way of asking your question is: given a (usually unbounded) self-adjoint operator $H$ on a Hilbert space $\mathscr{H}$, am I able to characterize its spectrum?

Finding "closed" solutions to the equation you are writing, means finding eigenfunctions of your operator $H$, possibly belonging to the Hilbert space $\mathscr{H}$ (since you want them to be realizable states). In other words it means that you investigate whether the operator $H$ has discrete spectrum.

Sometimes it is possible to prove that the spectrum is entirely discrete, sometimes that there is no discrete spectrum, but usually you have both discrete and non-discrete (essential) spectra: it really depends on the form of the operator $H$. There is an ongoing, huge and well established field of mathematical research on this subject called "spectral theory". The "bible" of mathematical physics, i.e. the books of Reed and Simon, dedicate an entire volume (the fourth) on this subject. I suggest you read the chapters VI, VII and VIII of the first volume as a general introduction, and whatever you like of Volume 4 (especially sections on bound states and eigenfunctions) to get an idea of the mathematical difficulties of analyzing the spectrum of operators, and consequently their eigenfunctions.

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