  Non-Hermitian systems can exhibit extraordinary boundary behaviors, known as
the non-Hermitian skin effects, where all the eigenstates are localized
exponentially at one side of lattice model. To give a full understanding and
control of non-Hermitian skin effects, we have developed the non-Hermitian
generalized Bloch theorem to provide the analytical expression for all solvable
eigenvalues and eigenstates, in which translation symmetry is broken due to the
open boundary condition. By introducing the Vieta's theorem for any polynomial
equation with arbitrary degree, our approach is widely applicable for
one-dimensional non-Hermitian tight-binding models. With the non-Hermitian
generalized Bloch theorem, we can analyze the condition of existence or
non-existence of the non-Hermitian skin effects at a mathematically rigorous
level. Additionally, the non-Hermitian generalized Bloch theorem allows us to
explore the real-space exceptional points. We also establish the connection
between our approach and the generalized Brillouin zone method. To illustrate
our main results, we examine two concrete examples including the
Su-Schrieffer-Heeger chain model with long-range couplings, and the ladder
model with non-reciprocal interaction. Our non-Hermitian generalized Bloch
theorem provides an efficient way to analytically study various non-Hermitian
phenomena in more general cases.
