October 2011
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Day October 21, 2011

New features of scattering from a one-dimensional non-Hermitian (complex) potential

Zafar Ahmed

For complex one-dimensional potentials, we propose the asymmetry of both reflectivity and transmitivity under time-reversal: \(R(-k)\ne R(k)$ and $T(-k) \ne T(k)\), unless the potentials are real or PT-symmetric. For complex PT-symmetry scattering potentials, we propose that \(R_{left}(-k)=R_{right}(k)\) and \(T(-k)=T(k)\).So far the spectral singularities (SS) of a one-dimensional non-Hermitian scattering potential are witnessed/conjectured to be at most one. We present a new non-Hermitian parametrization of Scarf II potential to reveal its four new features. Firstly, it displays the just acclaimed (in)variances. Secondly, it can support two spectral singularities at two pre-assigned real energies (\(E_*=\alpha^2,\beta^2\)) either in \(T(k)\) or in \(T(-k)\), when \(\alpha\beta>0\). Thirdly, when \(\alpha \beta <0\) it possesses one SS in \(T(k)\) and the other in \(T(-k)\). Lastly, when the potential becomes PT-symmetric \([(\alpha+\beta)=0]\), we get \(T(k)=T(-k)\), it possesses a unique SS at \(E=\alpha^2\) in both \(T(-k)\) and \(T(k)\).

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

Defect solitons supported by nonlocal PT symmetric superlattices

Sumei Hu, Daquan Lu, Xuekai Ma, Qi Guo, Wei Hu

The existence and stability of defect solitons supported by parity-time (PT) symmetric superlattices with nonlocal nonlinearity are investigated. Unlike local PT symmetric system, the nonlocal system considered reveals unusual properties. In the semi-infinite gap, in-phase solitons can exist stably for positive defects or zero defects, but can not exist for negative defects with the strong nonlocality. In the first gap, out-phase solitons are stable for positive defects or zero defects, whereas in-phase solitons are stable for negative defects. The dependence of soliton stabilities on modulation depth of the PT potentials is studied. It is interesting that solitons can exist stably for positive and zero defects when the PT potential is above the phase transition point.

Optics (physics.optics)

Symmetries of the Tachyonic Dirac Equation

U. D. Jentschura, B. J. Wundt

We show that it is possible to construct a tachyonic version of the Dirac equation, which contains the fifth current and reads (i gamma^\mu partial_\mu – gamma^5 m) \psi = 0. Its spectrum fulfills the dispersion relation E^2 = p^2 – m^2, where E is the energy, p is the spatial momentum, and m is the mass of the particle. The tachyonic Dirac equation is shown to be CP invariant, and T invariant. The Feynman propagator is found. In contrast to the covariant formulation, the tachyonic Hamiltonian H_5 = alpha.p + beta gamma^5 m breaks Lorentz covariance (as does the Hamiltonian formalism in general, because the time variable is singled out and treated differently from space). The tachyonic Dirac Hamiltonian H_5 breaks parity but is found to be invariant under the combined action of parity and a noncovariant time reversal operation T’. In contrast to the Lorentz-covariant T operation, T’ involves the Hermitian adjoint of the Hamiltonian. Thus, in the formalism developed by Bender et al., primarily in the context of quantum mechanics, H_5 is PT’ symmetric. The PT’ invariance (in the quantum mechanical sense) is responsible for the fact that the energy eigenvalues of the tachyonic Dirac Hamiltonian are real rather than complex. The eigenstates of the Hamiltonian are shown to approximate the helicity eigenstates of a Dirac neutrino in the massless limit.

High Energy Physics – Phenomenology (hep-ph)