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One-Dimensional Lieb-Oxford Bounds

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 $ρ_ψ$ 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 $I_\mathrm{xc}(ψ)\geq - C_1 \int_{\mathbb R} ρ_ψ(x)^{2} \mathrm{d}x$ is discussed for different convex potentials.