  Complex potential and non-Hermitian hopping amplitude are building blocks of
a non-Hermitian quantum network. Appropriate configuration, such as
PT-symmetric distribution, can lead to the full real spectrum. To investigate
the underlying mechanism of this phenomenon, we study the phase diagram of a
semi-infinite non-Hermitian system. It consists of a finite non-Hermitian
cluster and a semi-infinite lead. Based on the analysis of the solution of the
concrete systems, it is shown that it can have the full real spectrum without
any requirements on the symmetry and the wave function within the lead becomes
a unidirectional plane wave at the exceptional point. This universal dynamical
behavior is demonstrated as the persistent emission and reflectionless
absorption of wave packets in the typical non-Hermitian systems containing the
complex on-site potential and non-Hermitian hopping amplitude.
