A possible method for non-Hermitian and non-$PT$-symmetric Hamiltonian systems

摘要

A possible method to investigate non-Hermitian Hamiltonians is suggestedthrough finding a Hermitian operator $\eta_+$ and defining the annihilation andcreation operators to be $\eta_+$-pseudo-Hermitian adjoint to each other. Theoperator $\eta_+$ represents the $\eta_+$-pseudo-Hermiticity of Hamiltonians.As an example, a non-Hermitian and non-$PT$-symmetric Hamiltonian withimaginary linear coordinate and linear momentum terms is constructed andanalyzed in detail. The operator $\eta_+$ is found, based on which, a realspectrum and a positive-definite inner product, together with the probabilityexplanation of wave functions, the orthogonality of eigenstates, and theunitarity of time evolution, are obtained for the non-Hermitian andnon-$PT$-symmetric Hamiltonian. Moreover, this Hamiltonian turns out to becoupled when it is extended to the canonical noncommutative space withnoncommutative spatial coordinate operators and noncommutative momentumoperators as well. Our method is applicable to the coupled Hamiltonian. Thenthe first and second order noncommutative corrections of energy levels arecalculated, and in particular the reality of energy spectra, thepositive-definiteness of inner products, and the related properties (theprobability explanation of wave functions, the orthogonality of eigenstates,and the unitarity of time evolution) are found not to be altered by thenoncommutativity.