Investigation of non-Hermitian Hamiltonians in the Heisenberg picture

• Give the Heisenberg picture for non-Hermitian Hamiltonians.
• Construct two new non-Hermitian Hamiltonians with and without PT symmetry.
• Apply the similarity transformation to determine real eigenvalues of the two non-Hermitian Hamiltonians.
• Discuss the complementarity and compatibility between our proposal and the PT symmetry.

The Heisenberg picture for non-Hermitian but η-pseudo-Hermitian Hamiltonian systems is suggested. If a non-Hermitian but η-pseudo-Hermitian Hamiltonian leads to real second order equations of motion, though their first order Heisenberg equations of motion are complex, we can construct a Hermitian counterpart that gives the same second order equations of motion. In terms of a similarity transformation we verify the iso-spectral property of the Hermitian and non-Hermitian Hamiltonians and obtain the related eigenfunctions. This feature can be used to determine real eigenvalues for such non-Hermitian Hamiltonian systems. As an application, two new non-Hermitian Hamiltonians are constructed and investigated, where one is non-Hermitian and non-PT-symmetric and the other is non-Hermitian but PT-symmetric. Moreover, the complementarity and compatibility between our treatment and the PT symmetry are discussed.