The exchange-correlation potential experienced by an electron in the freespace adjacent to a solid surface or to a low-dimensional system defines thefundamental image states and is generally important in surface- andnano-science. Here we determine the potential near the two- and one-dimensionalelectron gases (EG), doing this analytically at the level of the exact exchangeof the density-functional theory (DFT). We find that, at $r_\perp\gg k_F^{-1}$,where $r_\perp$ is the distance from the EG and $k_F$ is the Fermi radius, thepotential obeys the already known asymptotic $-e^2/r_\perp$, while at $r_\perp\lesssim k_F^{-1}$, but {\em still in vacuum}, qualitative and quantitativedeviations of the exchange potential from the asymptotic law occur. Theplayground of the excitations to the low-lying image states falls into thelatter regime, causing significant departure from the Rydberg series. Ingeneral, our analytical exchange potentials establish benchmarks for numericalapproaches in the low-dimensional science, where DFT is by far the most commontool.
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