  The conventional bulk-boundary correspondence breaks down in non-Hermitian
systems. In this paper, we reestablish the bulk-boundary correspondence in
one-dimensional non-Hermitian systems by applying the scattering theory, which
is a systematical way in various symmetry classes. Based on the scattering
theory, it is discovered that the topological invariant can be obtained by
solving a generalized eigenproblem without calculating the generalized
Brillouin zone. As a direct consequence, we unveil a new type of topological
phase transition without typical bulk enengy gap closing and an unstable phase
with topological boundary states, dubbed the critical topological phase.
