Math Conjectures

Maximal ionization conjecture

OPENMajorConjectureProposed 1984 · Standard version

Canonical statement

For integers Z,N≥1Z,N\ge1, on the fermionic space ⋀NL2(R3;C2)\bigwedge^N L^2(\mathbb R^3;\mathbb C^2) let
HZ,N=∑i=1N(−12Δi−Z∣xi∣)+∑1≤i<j≤N1∣xi−xj∣,EZ(N)=inf⁡σ(HZ,N), H_{Z,N}=\sum_{i=1}^N\left(-\frac12\Delta_i-\frac Z{|x_i|}\right) +\sum_{1\le i<j\le N}\frac1{|x_i-x_j|}, \qquad E_Z(N)=\inf\sigma(H_{Z,N}),
and set EZ(0)=0E_Z(0)=0. There is a universal constant C<∞C<\infty such that EZ(N)<EZ(N−1)E_Z(N)<E_Z(N-1) implies N≤Z+CN\le Z+C for every Z,N≥1Z,N\ge1.
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For integers \(Z,N\ge1\), on the fermionic space \(\bigwedge^N L^2(\mathbb R^3;\mathbb C^2)\) let \[ H_{Z,N}=\sum_{i=1}^N\left(-\frac12\Delta_i-\frac Z{|x_i|}\right) +\sum_{1\le i<j\le N}\frac1{|x_i-x_j|}, \qquad E_Z(N)=\inf\sigma(H_{Z,N}), \] and set \(E_Z(0)=0\). There is a universal constant \(C<\infty\) such that \(E_Z(N)<E_Z(N-1)\) implies \(N\le Z+C\) for every \(Z,N\ge1\).
MATH-PHYS-004:[Lieb1984BoundMaximumNegativeIonization][Solovej2003IonizationHartreeFock][Nam2012NewBoundsIonization][Lewin2025ChargedQuantumParticles]
Mathematical physicsAnalysis and functional analysis

Notes

The maximal ionization conjecture asks how many electrons a nucleus can bind. For the nonrelativistic NN-electron Hamiltonian with a nucleus of charge ZZ, say the NN-th electron is bound when EZ(N)<EZ(N−1)E_Z(N)<E_Z(N-1); the conjecture asserts that this forces N≤Z+CN\le Z+C for a universal constant CC, matching the empirical fact that atoms carry at most a small excess negative charge. The question crystallized with Lieb's 1984 bound on the maximum negative ionization, which gives N<2Z+1N<2Z+1 in complete generality [Lieb1984BoundMaximumNegativeIonization].

In approximate theories the conjecture is settled: the excess charge is uniformly bounded in Thomas–Fermi theory, and Solovej proved the analogous statement in Hartree–Fock theory [Solovej2003IonizationHartreeFock]. For the full many-electron Schrödinger Hamiltonian, refined estimates have lowered the coefficient of ZZ below 22 [Nam2012NewBoundsIonization], but every known general bound still permits an excess charge that grows with ZZ, or carries a coefficient strictly above 11.

Proving a ZZ-independent bound on the excess charge for the true Coulomb Hamiltonian remains open; see [Lewin2025ChargedQuantumParticles] for a recent discussion of this and related problems.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.