Comment on `Supersymmetry, PT-symmetry and spectral bifurcation'

We demonstrate that the recent paper by Abhinav and Panigrahi entitled `Supersymmetry, PT-symmetry and spectral bifurcation' [Ann.\ Phys.\ 325 (2010) 1198], which considers two different types of superpotentials for the PT-symmetric complexified Scarf II potential, fails to take into account the invariance under the exchange of its coupling parameters. As a result, they miss the important point that for unbroken PT-symmetry this potential indeed has two series of real energy eigenvalues, to which one can associate two different superpotentials. This fact was first pointed out by the present authors during the study of complex potentials having a complex $sl(2)$ potential algebra.Comment: 6 pages, no figure, published versio

An update on PT-symmetric complexified Scarf II potential, spectral singularities and some remarks on the rationally-extended supersymmetric partners

The $\cal PT$-symmetric complexified Scarf II potential V(x)= - V_1 \sech^{2}x + {\rm i} V_2 \sech x \tanh x, $V_1>0$ , $V_{2}\neq 0$ is revisited to study the interplay among its coupling parameters. The existence of an isolated real and positive energy level that has been recently identified as a spectral singularity or zero-width resonance is here demonstrated through the behaviour of the corresponding wavefunctions and some property of the associated pseudo-norms is pointed out. We also construct four different rationally-extended supersymmetric partners to $V(x)$, which are $\cal PT$-symmetric or complex non-$\cal PT$-symmetric according to the coupling parameters range. A detailed study of one of these partners reveals that SUSY preserves the $V(x)$ spectral singularity existence.Comment: 14 pages, no figure, substantial additions on spectral singularities, title change

PT-symmetric non-polynomial oscillators and hyperbolic potential with two known real eigenvalues in a SUSY framework

Extending the supersymmetric method proposed by Tkachuk to the complex domain, we obtain general expressions for superpotentials allowing generation of quasi-exactly solvable PT-symmetric potentials with two known real eigenvalues (the ground state and first-excited state energies). We construct examples, namely those of complexified non-polynomial oscillators and of a complexified hyperbolic potential, to demonstrate how our scheme works in practice. For the former we provide a connection with the sl(2) method, illustrating the comparative advantages of the supersymmetric one.Comment: 14 pages, LaTeX, no figur

Non-Hermitian Hamiltonians with real and complex eigenvalues in a Lie-algebraic framework

We show that complex Lie algebras (in particular sl(2,C)) provide us with an elegant method for studying the transition from real to complex eigenvalues of a class of non-Hermitian Hamiltonians: complexified Scarf II, generalized P\"oschl-Teller, and Morse. The characterizations of these Hamiltonians under the so-called pseudo-Hermiticity are also discussed.Comment: LaTeX, 14 pages, no figure, 1 reference adde

Creation and annihilation operators and coherent states for the PT-symmetric oscillator

We construct two commuting sets of creation and annihilation operators for the PT-symmetric oscillator. We then build coherent states of the latter as eigenstates of such annihilation operators by employing a modified version of the normalization integral that is relevant to PT-symmetric systems. We show that the coherent states are normalizable only in the range (0, 1) of the underlying coupling parameter $\alpha$.Comment: one additional reference, final version to be published in MPL

A PT Symmetric QES Partner to the Khare Mandal Potential With Real Eigen Values

We consider a PT Symmetric Partner to Khare Mandal's recently proposed non-Hermitian potential with complex eigen values. Our potential is Quasi-Exactly solvable and is shown to possess only real eigen values.Comment: 10 page

PT-symmetric sextic potentials

The family of complex PT-symmetric sextic potentials is studied to show that for various cases the system is essentially quasi-solvable and possesses real, discrete energy eigenvalues. For a particular choice of parameters, we find that under supersymmetric transformations the underlying potential picks up a reflectionless part.Comment: 8 pages, LaTeX with amssym, no figure

PT-symmetric square well and the associated SUSY hierarchies

The PT-symmetric square well problem is considered in a SUSY framework. When the coupling strength $Z$ lies below the critical value $Z_0^{\rm (crit)}$ where PT symmetry becomes spontaneously broken, we find a hierarchy of SUSY partner potentials, depicting an unbroken SUSY situation and reducing to the family of $\sec^2$-like potentials in the $Z \to 0$ limit. For $Z$ above $Z_0^{\rm (crit)}$, there is a rich diversity of SUSY hierarchies, including some with PT-symmetry breaking and some with partial PT-symmetry restoration.Comment: LaTeX, 18 pages, no figure; broken PT-symmetry case added (Sec. 6

Generalized Continuity Equation and Modified Normalization in PT-Symmetric Quantum Mechanics

The continuity equation relating the change in time of the position probability density to the gradient of the probability current density is generalized to PT-symmetric quantum mechanics. The normalization condition of eigenfunctions is modified in accordance with this new conservation law and illustrated with some detailed examples.Comment: 16 pages, amssy