{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"554f023c8080f383f1c85b3d8c18a0b55385b3cab056d7a9bbb64927802e47fe","created":"2026-10-03T07:18:06Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"cf4a25250d0a2fc42debb4f03abdc355ad6691892bc54cf89cf1b95e098bcc58","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"mathph.ionization.ionization-energy","kind":"well-posed","literature_status":"open","n":"1","parents":[],"plain":"The energy needed to remove one electron from a neutral atom is a few eV for every element in the periodic table. Prove that it stays bounded as Z grows.","posed_since":"","precise":"Let $I(Z) = E(Z-1,Z) - E(Z,Z)$ for the Hamiltonian $H_{N,Z}$ of mathph.ionization.excess-charge-bound. Prove $\\operatorname{sup}_Z I(Z) < \\infty$, i.e. $I(Z) \\le C m e^4/\\hbar^2$ with $C$ independent of $Z$, or show $I(Z)$ is unbounded.","problem_ref":null,"references":"","settled_by":"A Z-independent upper bound on E(Z-1,Z) - E(Z,Z) derived for the full many-electron Hamiltonian.","status_note":"Proved in Hartree-Fock theory (Solovej 2003); for the Schrodinger Hamiltonian the best known bound grows with $Z, I(Z) \\le C Z^{5/7}$ (Seco, Sigal, Solovej 1990).","title":"Uniform bound on the ionization energy of neutral 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