Pseudo-reality and pseudo-adjointness of Hamiltonians

We define pseudo-reality and pseudo-adjointness of a Hamiltonian, H, as ρHρ−1 = H∗ and μHμ−1 = H ′, respectively. We prove that the former yields the necessary condition for spectrum to be real whereas the latter helps in fixing a definition for inner-product of the eigenstates. Here we separate out adjointness of an operator from its Hermitian-adjointness. It turns out that a Hamiltonian possessing real spectrum is first pseudo-real, further it could be Hermitian, PT-symmetric or pseudo-Hermitian. Last few years have witnessed a new scope for even a non-Hermitian Hamiltonians to possess real spectrum. It has been found that when a Hamiltonian, H , is invariant under the joint action of parity (P : x → −x) and time-reversal (T : i → −i) i.e. [PT, H] = 0, there may arise surprisingly two situations. One, when PT and H admit common eigenstates and two, when they do not do so. In the former situation one can show that the eigenvalues will be real and in the latter the eigenvalues are conjectured to be complex-conjugate pairs and PT-symmetry is said to be spontaneously broken. One can indeed not tell whether a PT -symmetric potential has real or complex (conjugate pairs of) energy-eigenvalues, until the wavefunctions are analyzed. This intriguing feature has inspired the pursuit [1-6] of both analytically and numerically solved models of PT-symmetric potentials. Thus, PTsymmetry of a Hamiltonian could at most be a necessary and not the sufficient condition for the reality of eigenvalues. Further, the phenomenon of real eigenvalues of non-Hermitian Hamiltonians has been found to be connected with the already known concept of pseudo-Hermiticity. A Hamiltonian is pseudo-Hermitian [7-16] if