Two non-Hermitian PT-symmetric Hamiltonian systems are reconsidered by means of the algebraic method which was originally proposed for the pseudo-Hermitian Hamiltonian systems rather than for the PT-symmetric ones. Compared with the way converting a non-Hermitian Hamiltonian to its Hermitian counterpart, this method has the merit that keeps the Hilbert space of the non-Hermitian PT-symmetric Hamiltonian unchanged. In order to give the positive definite inner product for the PT-symmetric systems, a new operator V, instead of C, can be introduced. The operator V has the similar function to the operator C adopted normally in the PT-symmetric quantum mechanics, however, it has the advantage that V can be constructed directly in terms of Hamiltonians. The spectra of the two non-Hermitian PT-symmetric systems are obtained, which coincide with that given in literature, and in particular, the Hilbert spaces associated with positive definite inner products are worked out.

http://arxiv.org/abs/1204.6544
Quantum Physics (quant-ph); High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph)

By adding an imaginary potential proportional to \(ip_1p_2\) to the hamiltonian of an anisotropic planar oscillator, we construct a model which is described by a non-hermitian hamiltonian with PT pseudo-hermiticity. We introduce the mechanism of the spontaneous breaking of permutation symmetry of the hamiltonian for diagonalizing the hamiltonian. By applying the definition of annihilation and creation operators which are PT pseudo-hermitian adjoint to each other, we give the real spectra.

http://arxiv.org/abs/1110.2312
Quantum Physics (quant-ph)

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)