On the Dirac and Schwinger corrections to the ground-state energy of an atom. (English) Zbl 0820.35113

Fefferman, C.; Seco, L. A.

Adv. Math. 107, No. 1, 1-185 (1994).

Let \(E(N,q,Z, \text{spin})\) be the infimum of the spectrum of the Hamiltonian \(H\) for \(N\) electrons and a nucleus of charge \(Z\) acting on a spin Hilbert space. \(E(q,Z)\) is the infimum of \(E(N,q,Z, \text{spin})\) over all choices of \(N\) and spin from \(q\) possible values. The authors give here the following formula: \[ E(q,Z) = - c_ 0 q^{2/3} Z^{7/3} + qZ^ 2/8 - (1 + q/9) q^{1/3} c_ 1 Z^{5/3} + \text{ERROR} (Z,q), \]
\[ \bigl | \text{ERROR} (Z,q) \bigr | \leq C_ q Z^{5/3 - 1/2835}. \] That is, they give upper and lower bounds for \(E(q,Z)\) that differ by \(O(Z^{5/3})\). The upper bound is proved by computing the energy of a Hartree-Fock trial wave function. To prove the lower bound, they break up the Coulomb repulsion \(\Sigma_{i<j} | x_ i - x_ j |^{-1}\) into a long-range part and a short-range part. Important part of their proof is the treatment of the short-range part.
Finally they show the results on quantum states \(\Psi\) for an atom having nearly the lowest possible energy. They also conjecture a sharp formula for the sum of the negative eigenvalues of \(- \Delta + V_{TF}\), where \(V_{TF}\) is the Thomas-Fermi potential. Some parts of their proof can go over for molecules.

Reviewer: H.Yamagata (Osaka)

MSC:

35Q40	PDEs in connection with quantum mechanics
81V45	Atomic physics
35P15	Estimates of eigenvalues in context of PDEs

Keywords:

spectrum of the Hamiltonian; upper and lower bounds; wave function; Coulomb repulsion