The eigenvalue and eigenvector of a non-Hermitian Hamiltonian

The non-Hermitian Hamiltonian is constructed with the knot operators, and then the eigenvalue and corresponding eigenvector are evaluated respectively． It manifests that the eigenvalue of a non-Hermitian Hamiltonian is a complex number, and changes with the angle and the tunable parameter． The number and position of exceptional points are obtained theoretically． Moreover, the biorthogonal normalization of the right and left eigenvectors of the non-Hermitian Hamiltonian is discussed, which is different from the case in the traditional quantum mechanics． Finally, according to the Kirchhoff’s current law, the non-Hermitian Hamiltonian is realized experimentally in an electric circuit with resistor, inductor and capacitor components．