v-Representability and Density Functional Theory

It is shown that if $n\left(r\right)\mathrm{}$ is the discrete density on a lattice (enclosed in a finite box) associated with a nondegenerate ground state in an external potential $v\left(r\right)\mathrm{}$ (i.e., is "$v$-representable"), then the density $n\left(r\right)+\mu m\left(r\right)\mathrm{}$, with $m\left(r\right)\mathrm{}$ arbitrary (apart from trivial constraints) and $\mu$ small enough, is also associated with a nondegenerate ground state in an external potential $v^{\prime }\mathrm{}\left(r\right)\mathrm{}$ near $v\left(r\right)\mathrm{}$; i.e., $n\left(r\right)+\mu m\left(r\right)\mathrm{}$ is also $v$-representable. Implications for the Hohenberg-Kohn variational principle and the Kohn-Sham equations are discussed.