  The energy bands of non-Hermitian systems exhibit nontrivial topological
features that arise from the complex nature of the energy spectrum. Under
periodic boundary conditions (PBC), the energy spectrum describes rather
generally closed loops in complex plane, characterized by integer nonzero
winding numbers. Such nontrivial winding provides the topological signature of
the non-Hermitian skin effect (NHSE), i.e. the macroscopic condensation of bulk
states at the lattice edges under open boundary conditions (OBC). In spite of
the great relevance of band winding in the non-Hermitian topological band
theory and the related NHSE, most of current results rely on tight-binding
models of non-Hermitian systems, while exact Bloch wave function analysis of
the NHSE and related topological band theory is still lacking. While
tight-binding models can correctly describe narrow-band electronic states with
a relatively weak degree non-Hermiticity, they are not suited to describe
high-energy wide-band electronic states and/or regimes corresponding to strong
non-Hermiticity. Here we consider the single-particle continuous Schr\"odinger
equation in a periodic potential, in which non-Hermiticity is introduced by an
imaginary vector potential in the equation, and show that the NHSE is
ubiquitous under OBC and characterized by a non-vanishing integer winding
number, even thought the energy spectrum under PBC always comprises an open
curve, corresponding to high-energy electronic states. We also show that the
interior of the PBC energy spectrum corresponds to the complex eigenenergies
sustaining localized (edge) states under semi-infinite boundary conditions.
