  Non-Hermitian skin effect (NHSE) is a unique phenomenon studied intensively
in non-Hermitian systems during the past few years. In this work, we discuss
the energy dependence of NHSE by introducing nonreciprocity beyond the
nearest-neighboring hopping in the one-dimensional lattices. The direction of
NHSE reverses as the eigenenergy of the system under open boundary conditions
(OBCs) sweeps across some critical energies. To characterize such a phenomenon,
we introduce the concept of non-Hermitian skin effect edges, which separate the
eigenstates localized at opposite ends of the lattice in the OBC spectrum. We
find the skin effect edges are determined by the self-intersections in the
spectrum under periodic boundary conditions (PBCs), which are topological as
the winding number of the PBC spectrum changes sign when crossing them.
Moreover, the NHSE edges will disappear when the self-intersections merge into
a single point by tuning the system parameters. Our work reveals the intricate
interplay between NHSE and nonreciprocal hopping in non-Hermitian systems.
