An algebraic method for pseudo-hermitian systems is proposed through redefining annihilation and creation operators which are pseudo-hermitian adjoint to each other. As an example, a parity-pseudo-hermitian Hamiltonian is constructed and then analyzed in detail. Its real spectrum is obtained by means of the algebraic method, in which a new operator $V$ is introduced in order to define new annihilation and creation operators and to keep pseudo-hermitian inner products positive definite. It is shown that this P-pseudo-hermitian Hamiltonian also possesses PV-pseudo-hermiticity, where PV ensures a positive definite inner product. Moreover, when the parity-pseudo-hermitian system is extended to the canonical noncommutative space with noncommutative spatial coordinates and noncommutative momenta as well, the first order noncommutative correction of energy levels is calculated, and in particular the reality of energy spectra and the positive definiteness of inner products are found to be not altered by the noncommutativity.

http://arxiv.org/abs/1107.4972
Quantum Physics (quant-ph); High Energy Physics – Theory (hep-th)