One-dimensional Lieb–Oxford bounds

Abstract

We investigate and prove Lieb–Oxford bounds in one dimension by studying convex potentials that approximate the ill-defined Coulomb potential. A Lieb–Oxford inequality establishes a bound of the indirect interaction energy for electrons in terms of the one-body particle density ρψ of a wave function ψ. Our results include modified soft Coulomb potential and regularized Coulomb potential. For these potentials, we establish Lieb–Oxford-type bounds utilizing logarithmic expressions of the particle density. Furthermore, a previous conjectured form Ixc(ψ)≥−C1∫ℝρψ(x)2dx is discussed for different convex potentials.