{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"9281a8e851779c8a05c1b2fc6d511a1573b87b99c7eacd3baf3c151080ce3eef","created":"2026-10-03T07:18:06Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"c5705f6ed2251c849c1bee8630ae81e6c0d73de71ac93b4e4e03ebde4a019d4a","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"mathph.ionization.excess-charge-bound","kind":"well-posed","literature_status":"open","n":"1","parents":[],"plain":"Show that there is one fixed number C such that no atom, however heavy, can bind more than Z + C electrons. Bind means the extra electrons stay attached instead of flying off to infinity.","posed_since":"1984","precise":"For $H_{N,Z} = \\sum_{i=1}^N (-\\hbar^2/(2m) \\Delta_i - Z e^2/|x_i|) + \\sum_{i<j} e^2/|x_i - x_j|$ on antisymmetric $L^2((R^3 x \\{\\mathrm{up},\\mathrm{down}\\})^N)$ with a static point nucleus, let $E(N,Z)$ be its ground-state energy and $N_c(Z)$ the largest N with $E(N,Z) < E(N-1,Z)$. Prove or disprove $N_c(Z) \\le Z + C$ with C independent of Z. Answer: a proof with explicit or existential C, or a sequence $Z_k$ with $N_c(Z_k) - Z_k$ unbounded.","problem_ref":null,"references":"","settled_by":"A proof that bounds $N_{c}(Z) - Z$ uniformly in $Z$, for example by controlling the screened nuclear potential felt by the outermost electrons in the many-body ground state.","status_note":"Best known: $N_{c} < 2Z + 1$ (Lieb 1984), $N_{c} < 1.22 Z + 3 Z^{1/3}$ (Nam 2012), $N_{c}/Z \\to 1$ (Lieb, Sigal, Simon, Thirring 1988) and $N_{c} - Z \\le C Z^{5/7}$ for large Z (Fefferman, Seco 1990; Seco, Sigal, Solovej 1990); the bounded-excess statement is proved in Hartree-Fock theory (Solovej, Annals 2003) and in Thomas-Fermi-Dirac-von Weizsacker theory (Frank, Nam, Van Den Bosch, CPAM 2018), not for the Schrodinger Hamiltonian.","title":"Excess electron number of atoms bounded independently of nuclear 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