  We explore band structures of one-dimensional open systems described by
periodic non-Hermitian operators, based on continuum models and tight-binding
models. We show that imaginary scalar potentials do not open band gaps but
instead lead to the formation of exceptional points as long as the strength of
the potential exceeds a threshold value, which is contrast to closed systems
where real potentials open a gap with infinitesimally small strength. The
imaginary vector potentials hinder the separation of low energy bands because
of the lifting of degeneracy in the free system. In addition, we construct
tight-binding models through bi-orthogonal Wannier functions based on Bloch
wavefunctions of the non-Hermitian operator and its Hermitian conjugate. We
show that the bi-orthogonal tight-binding model well reproduces the dispersion
relations of the continuum model when the complex scalar potential is
sufficiently large.
