Jun-Qing Li, Yan-Gang Miao, Zhao Xue
An algebraic method for pseudo-Hermitian Hamiltonian systems is proposed through introducing the operator $\eta_+$ and the $\eta_+$-pseudo-Hermitian adjoint of states and then redefining annihilation and creation operators to be $\eta_+$-pseudo-Hermitian (not Hermitian) adjoint of each other. As an example, a parity-pseudo-Hermitian Hamiltonian is constructed and analyzed in detail. Its real spectrum is obtained by means of the algebraic method, where the corresponding operator $\eta_+$ is found to be $PV$ through a specific choice of $V$. The operator $V$ is given in such a way that on the one hand this $P$-pseudo-Hermitian Hamiltonian is also $PV$-pseudo-Hermitian self-adjoint and on the other hand $PV$ ensures a real spectrum and a positive-definite inner product. Moreover, when the $P$-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.
View original:
http://arxiv.org/abs/1107.4972
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