A physical interpretation for the non-Hermitian Hamiltonian

We explore a way of finding the link between a non-Hermitian Hamiltonian and a Hermitian one. Based on the analysis of Bethe ansatz solutions for a class of non-Hermitian Hamiltonians and the scattering problems for the corresponding Hermitian Hamiltonians, it is shown that a scattering state of an arbitrary Hermitian lattice embedded in a chain as the scattering center shares the same wavefunction with the corresponding non-Hermitian tight binding lattice, which consists of the Hermitian lattice with two additional on-site complex potentials, no matter whether the non-Hermitian is broken PT symmetry or even non-PT symmetry. Exactly solvable models are presented to demonstrate the main points of this paper.