Zafar Ahmed, Amal Nathan Joseph

Hitherto, it is well known that complex PT-symmetric Scarf II has real discrete spectrum in the parametric domain of unbroken PT-symmetry. We reveal new interesting complex, non-PT-symmetric parametric domains of this versatile potential, \(V(x)\), where the spectrum is again discrete and real. Showing that the Hamiltonian, \(p^2/2m+V(x)\), is pseudo-Hermitian could be challenging, if possible.

http://arxiv.org/abs/1411.3231

Quantum Physics (quant-ph); Mathematical Physics (math-ph)

Dmitry E. Pelinovsky, Dmitry A. Zezyulin, Vladimir V. Konotop

We study a system of two coupled nonlinear Schrodinger equations, where one equation includes gain and the other one includes losses. Strengths of the gain and the loss are equal, i.e., the resulting system is parity-time (PT) symmetric. The model includes both linear and nonlinear couplings, such that when all nonlinear coefficients are equal, the system represents the PT-generalization of the Manakov model. In the one-dimensional case, we prove the existence of a global solution to the Cauchy problem in energy space \(H_1\), such that the \(H_1\)-norm of the global solution may grow in time. In the Manakov case, we show analytically that the \(L_2\)-norm of the global solution is bounded for all times and numerically that the \(H_1\)-norm is also bounded. In the two-dimensional case, we obtain a constraint on the \(L_2\)-norm of the initial data that ensures the existence of a global solution in the energy space \(H_1\).

http://arxiv.org/abs/1411.2895

Analysis of PDEs (math.AP)