Jellium Energy

The jellium energy is a variational energy for points interacting through the Coulomb potential in a uniform compensating background. For a bounded open region , background density , and points , ..., , the classical energy is

(1)

The three terms describe pair interaction, interaction with the background, and the self-interaction of the background.

For the quantum variational problem, require to be a positive integer and fix an integer spin multiplicity . The Coulomb Hamiltonian is

(2)

where and is the Laplacian. The ground state energy is

(3)

where depends on position-spin pairs with and and is antisymmetric under exchanging any two such pairs. The infimum is over the Dirichlet quadratic-form domain , which imposes homogeneous Dirichlet boundary conditions. Here is volume and is the L²-norm. For growing balls with , the limit

(4)

exists. The same value is obtained for domain sequences satisfying the regularity hypotheses of Lieb and Narnhofer (1975).

The uniform electron gas imposes a different variational constraint. Its one-particle density is prescribed to be uniform away from a smoothed boundary, while the particle number may fluctuate. Its energy minimizes the kinetic and pair-interaction energies and subtracts the direct Coulomb self-energy of that prescribed density. The thermodynamic limit of this minimum per unit volume defines (Lewin et al. 2018). Jellium fixes the background density but does not require the particle density to be uniform.

Csirik (2026) reported

(5)

The claim applies in three dimensions for every , , and integer spin multiplicity . Csirik reports substantive ChatGPT and Claude assistance with proof development and formalization. The Lean development checks comparison arguments and a neutralization construction relative to hypotheses on abstract states and charges. The thermodynamic-limit results and the Hilbert space representation and energy identities remain inputs. Independent specialist review had not been reported as of Oct. 25, 2026.