  Non-Hermitian skin effect, namely that the eigenvalues and eigenstates of a
non-Hermitian tight-binding Hamiltonian have significant differences under open
or periodic boundary conditions, is a remarkable phenomenon of non-Hermitian
systems. Inspired by the presence of the non-Hermitian skin effect, we study
the evolution of wave-packets in non-Hermitian systems, which can be determined
using the single-particle Green's function. Surprisingly, we find that in the
thermodynamical limit, the Green's function does not depend on boundary
conditions, despite the presence of skin effect. We proffer a general proof for
this statement in arbitrary dimension with finite hopping range, with an
explicit illustration in the non-Hermitian Su-Schrieffer-Heeger model. We also
explore its applications in non-interacting open quantum systems described by
the master equation, where we demonstrate that the evolution of the density
matrix is independent of the boundary condition.
