Frequency-dependentv-representability in density-functional theory

In density-functional theory (DFT) of the ground state, a density distribution $n_o$(r) is called v-representable (VR) if it is the ground-state density in some external potential. [It is known that not all ‘‘reasonable’’ $n_0$(r) are VR.] In DFT of time-dependent linear response of a nondegenerate ground state, a similar question arises: Is a response density, $n_1$(r,ω)$e^{\mathrm{-}i\omega t}$, VR, i.e., is it the response to some perturbing potential $v_1$(r,ω)$e^{\mathrm{-}i\omega t}$ [E.K.U. Gross and W. Kohn, Phys. Rev. Lett. 55, 2850 (1985)]? In the present paper we show that (1) if the frequency ω<${\omega }_{\min }$ (the lowest excitation frequency), the answer is affirmative; and (2) if ω>${\omega }_{\min }$, the answer is not necessarily affirmative, as demonstrated by counterexamples. [We exhibit ‘‘reasonable’’ functions $n_1$(r,ω)$e^{\mathrm{-}i\omega t}$ which, at isolated frequencies, are not VR.] Implications for time-dependent DFT of linear response are discussed.