The Hohenberg–Kohn Theorem

Luca Salasnich

Viraht Sahni, Xiao‐Yin Pan

V. Mikoláš, Mojmír Tomášek

Paul E. Lammert

Thiago Carvalho Corso

We show that for Schrödinger operators in a connected domain, the Hohenberg-Kohn theorem holds within the class of Laplace form-bounded external potentials if and only if the single-particle density is strictly positive quasi-everywhere. Furthermore, we show that this condition is satisfied for non-interacting Schrödinger operators whenever a ground-state exists. Consequently, we establish the Hohenberg-Kohn theorem for non-interacting systems, and thereby the uniqueness of the Kohn-Sham potential, within the maximal class of Laplace form-bounded distributions. The main ingredient to establish these results is a characterization of regular states, whose proof relies on tools from classical potential theory. Moreover, this characterization reveals that, in the continuum setting, the fundamental mechanism underlying the Hohenberg-Kohn theorem is the (quasi)-unique continuation of the density rather than of the many-body wavefunction.

Jacob Katriel

Theophilou's extension (ibid., vol.12, p.5419, 1979) of the Hohenberg-Kohn theorem to excited states is shown to follow from specialisation of the familiar Hohenberg-Kohn theorem to the ground state of an appropriately defined non-interacting super-Hamiltonian.