For the hydrogen atom, a simple separation of variables give the energy eigenvalue of the Schrodinger operator for one electron in a spherical potential. It is well known that there are no such explicit solutions for the helium atom. However, is there a rigorous mathematical proof that the helium atom's Hamiltonian operator indeed has discretized eigenvalues?
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Qmechanic
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Simplyorange
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1This very close to, and possibly a duplicate of, How to prove bound state's spectrum must be discrete and scattering state's spectrum must be continuous? – John Rennie May 01 '23 at 05:02
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1It sounds to me like OP is asking for an existence proof for bound states, which is not addressed in the post you mention. https://link.springer.com/article/10.1007/BF02730332 discusses this question in a simplified setting with no spin – Prox May 08 '23 at 15:34