Frequency-dependentv-representability in density-functional theory

In density-functional theory (DFT) of the ground state, a density distribution nₒ(r) is called v-representable (VR) if it is the ground-state density in some external potential. [It is known that not all ``reasonable'' n₀(r) are VR.] In DFT of time-dependent linear response of a nondegenerate ground state, a similar question arises: Is a response density, n₁(r,{ω})e^-iωt, VR, i.e., is it the response to some perturbing potential v₁(r,{ω})e^-iωt [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}ωₘᵢₙ (the lowest excitation frequency), the answer is affirmative; and (2) if {ω}>ωₘᵢₙ, the answer is not necessarily affirmative, as demonstrated by counterexamples. [We exhibit ``reasonable'' functions n₁(r,{ω})e^-iωt which, at isolated frequencies, are not VR.] Implications for time-dependent DFT of linear response are discussed.