  In the Hartree-Fock (HF) theory, the Koopmans' theorem stipulates that the
first ionization potential (IP) is equal to minus the energy of the highest
occupied molecular orbital. A similar result, known as the IP-theorem, holds in
the Kohn-Sham (KS) density functional theory (DFT). We argue that proofs of
these theorems rely critically on the finiteness of the number of particles
comprising the considered systems. We revisit the problem with focus on
extended systems and we confirm the Koopmans' theorem validity. On the other
hand, we establish the IP-theorem's violation within the context of KS DFT of
extended systems. These findings have an immediate bearing on the role of the
KS valence band at solid surfaces, mesoscopic, and other extended systems with
respect to the work-function problem. Based on our results, we introduce a
concept of the ionization potential band structure as an observable alternative
to its unphysical KS counterpart. A method of practical determination of IP
band structure in terms of DFT quantities is provided.
