  Although the non-Bloch band theory is a milestone in elaborating bulk energy
bands of non-Hermitian systems under the open-boundary condition (OBC), vital
issues related to multivalued functions of non-Hermitian energy bands remain
unsolved. In this paper, we anatomize the bulk properties of one-dimensional
multiband non-Hermitian systems under OBC. We put forward the energy-band
branches (EBBs) to settle the multivalued functions of non-Hermitian energy
bands, which become gapped or gapless corresponding to disconnected or
connected EBBs in the complex energy plane, where the branch points and branch
cuts play a crucial role. We clarify the precise significance of the
non-Hermitian skin effect, which illustrates the asymptotic behavior of EBB
eigenstates (bulk eigenstates) in the deep bulk and compensates previous
non-Bloch band theory. We also obtain a general expression of open-boundary
Green's functions based on such EBBs and generalized Brillouin zones, useful
for studies on non-Hermitian dynamical evolution.
