Existence of Hartree-Fock excited states for atoms and molecules

Mathieu Lewin (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

For neutral and positively charged atoms and molecules, we prove the existence of infinitely many Hartree-Fock critical points below the first energy threshold (that is, the lowest energy of the same system with one electron removed). This is the equivalent, in Hartree-Fock theory, of the famous Zhislin-Sigalov theorem which states the existence of infinitely many eigenvalues below the bottom of the essential spectrum of the $N$-particle linear Schrödinger operator. Our result improves a theorem of Lions in 1987 who already constructed infinitely many Hartree-Fock critical points, but with much higher energy. Our main contribution is the proof that the Hartree-Fock functional satisfies the Palais-Smale property below the first energy threshold. We then use minimax methods in the $N$-particle space, instead of working in the one-particle space.

Domaines: Physique mathématique [math-ph]; Equations aux dérivées partielles [math.AP]; Théorie spectrale [math.SP]

Mathieu Lewin. Existence of Hartree-Fock excited states for atoms and molecules. Letters in Mathematical Physics, 2018, 108 (4), pp.985-1006. ⟨10.1007/s11005-017-1019-y⟩. ⟨hal-01570624v3⟩