  The IP-theorem of the Kohn-Sham (KS) density functional theory (DFT) states
that the energy of the highest occupied molecular orbital (HOMO)
$\epsilon_{HOMO}$ equals the negative of the first ionization potential (IP),
thus ascribing a physical meaning to one of the eigenvalues of the KS
hamiltonian. We scrutinize the fact that the validity of the IP-theorem relies
critically on the electron density $n({\bf r})$, far from the system, to be
determined by HOMO only, behaving as $n({\bf r}) \underset{r\to\infty}{\sim}
e^{- 2 \sqrt{-2 \epsilon_{HOMO}} r}$. While this behavior always holds for
finite systems, it does not hold for mesoscopic ones, such as
quasi-two-dimensional (Q2D) electron gas or Q2D crystals. We show that this
leads to the violation of the IP-theorem for the latter class of systems. This
finding has a strong bearing on the role of the KS valence band with respect to
the work-function problem in the mesoscopic case. Based on our results, we
introduce a concept of the IP band structure as an observable alternative to
its unphysical KS counterpart. A practical method of the determination of IP
band structure in terms of DFT quantities is provided.
