## Tracking down localized modes in PT-symmetric Hamiltonians under the influence of a competing nonlinearity

Bijan Bagchi, Subhrajit Modak, Prasanta K. Panigrahi

The relevance of parity and time reversal (PT)-symmetric structures in optical systems is known for sometime with the correspondence existing between the Schrodinger equation and the paraxial equation of diffraction where the time parameter represents the propagating distance and the refractive index acts as the complex potential. In this paper, we systematically analyze a normalized form of the nonlinear Schrodinger system with two new families of PT-symmetric potentials in the presence of competing nonlinearities. We generate a class of localized eigenmodes and carry out a linear stability analysis on the solutions. In particular, we find an interesting feature of bifurcation charaterized by the parameter of perturbative growth rate passing through zero where a transition to imaginary eigenvalues occurs.

http://arxiv.org/abs/1307.7246
Quantum Physics (quant-ph); Mathematical Physics (math-ph); Exactly Solvable and Integrable Systems (nlin.SI)

## Conserved Correlation in PT -symmetric Systems: Scattering and Bound States

Kumar Abhinav, Arun Jayannavar, P. K. Panigrahi

For one-dimensional PT -symmetric systems, it is observed that the non-local product obtained from the continuity equation can be interpreted as a conserved corre- lation function. This leads to physical conclusions, regarding both discrete and continuum states of such systems. Asymptotic states are shown to have necessarily broken PT -symmetry, leading to modified scattering and transfer matrices. This yields restricted boundary conditions, e.g., in- cidence from both sides, analogous to that of the proposed PT CPA laser. The interpretation of left and right states leads to a Hermitian S-matrix, resulting in the non-conservation of the flux. This further satisfies a duality condition, identical to the optical analogues. However, the non-local conserved scalar implements alternate boundary conditions in terms of in and out states, leading to the pseudo-Hermiticity condition in terms of the scattering matrix. Interestingly, when PT -symmetry is preserved, it leads to stationary states with real energy, naturally inter- pretable as bound states. The broken PT -symmetric phase is also captured by this correlation, with complex-conjugate pair of energies, interpreted as resonances.

http://arxiv.org/abs/1109.3113
Quantum Physics (quant-ph)

## Comment on “Comment on ‘Supersymmetry, PT-symmetry and spectral bifurcation’”

Kumar Abhinav, Prasanta K. Panigrahi

In “Comment on Supersymmetry, PT-symmetry and spectral bifurcation” \cite{BQ1}, Bagchi and Quesne correctly show the presence of a class of states for the complex Scarf-II potential in the unbroken PT-symmetry regime, which were absent in \cite{AP}. However, in the spontaneously broken PT-symmetry case, their argument is incorrect since it fails to implement the condition for the potential to be PT-symmetric: $C^{PT}[2(A-B)+\alpha]=0$. It needs to be emphasized that in the models considered in \cite{AP}, PT is spontaneously broken, implying that the potential is PT- symmetric, whereas the ground state is not. Furthermore, our supersymmetry (SUSY)-based ‘spectral bifurcation’ holds \textit{independent} of the $sl(2)$ symmetry consideration for a large class of PT-symmetric potentials.

http://arxiv.org/abs/1010.1909
Quantum Physics (quant-ph)