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Month June 2014

\(\mathcal{PT}\)-symmetric Hamiltonian Model and Exactly Solvable Potentials

Özlem Yeşiltaş

Searching for non-Hermitian (parity-time)\(\mathcal{PT}\)-symmetric Hamiltonians with real spectra has been acquiring much interest for fourteen years. In this article, we have introduced a \(\mathcal{PT}\)-symmetric non-Hermitian Hamiltonian model which is given as \(\hat{\mathcal{H}}=\omega (\hat{b}^\dagger\hat{b}+\frac{1}{2})+\alpha (\hat{b}^{2}-(\hat{b}^\dagger)^{2})\) where \(\omega\) and \(\alpha\) are real constants, \(\hat{b}\) and \(\hat{b^\dagger}\) are first order differential operators. Moreover, Pseudo-Hermiticity that is a generalization of \(\mathcal{PT}\) symmetry has been attracting a growing interest \cite{mos}. Because the Hamiltonian \(\mathcal{H}\) is pseudo-Hermitian, we have obtained the Hermitian equivalent of \(\mathcal{H}\), which is in Sturm- Liouville form, leads to exactly solvable potential models which are effective screened potential and hyperbolic Rosen-Morse II potential. \(\mathcal{H}\) is called pseudo-Hermitian, if there exists a Hermitian and invertible operator \(\eta\) satisfying \(\mathcal{H^\dagger}=\eta \mathcal{H} \eta^{-1}\). For the Hermitian Hamiltonian \(h\), one can write \(h=\rho \mathcal{H} \rho^{-1}\) where \(\rho=\sqrt{\eta}\) is unitary. Using this \(\rho\) we have obtained a physical Hamiltonian \(h\) for each case. Then, the Schr\”{o}dinger equation is solved exactly using Shape Invariance method of Supersymmetric Quantum Mechanics. Mapping function \(\rho\) is obtained for each potential case.


Quantum Physics (quant-ph)

Metric Operator For The Non-Hermitian Hamiltonian Model and Pseudo-Supersymmetry

Özlem Yeşiltaş, Nafiye Kaplan

We have obtained the metric operator \(\Theta=\exp T\) for the non-Hermitian Hamiltonian model \(H=\omega(a^{\dag}a+1/2)+\alpha(a^{2}-a^{\dag^{2}})\). We have also found the intertwining operator which connects the Hamiltonian to the adjoint of its pseudo-supersymmetric partner Hamiltonian for the model of hyperbolic Rosen-Morse II potential.

Mathematical Physics (math-ph)

PT-symmetric sine-Gordon breathers

N. Lu, J. Cuevas-Maraver, P.G. Kevrekidis

In this work, we explore a prototypical example of a genuine continuum breather (i.e., not a standing wave) and the conditions under which it can persist in a \(\mathcal{P T}\)-symmetric medium. As our model of interest, we will explore the sine-Gordon equation in the presence of a \(\mathcal{P T}\)- symmetric perturbation. Our main finding is that the breather of the sine-Gordon model will only persist at the interface between gain and loss that \(\mathcal{P T}\)-symmetry imposes but will not be preserved if centered at the lossy or at the gain side. The latter dynamics is found to be interesting in its own right giving rise to kink-antikink pairs on the gain side and complete decay of the breather on the lossy side. Lastly, the stability of the breathers centered at the interface is studied. As may be anticipated on the basis of their “delicate” existence properties such breathers are found to be destabilized through a Hopf bifurcation in the corresponding Floquet analysis.

Pattern Formation and Solitons (nlin.PS)

Exact Solutions for Non-Hermitian Dirac-Pauli Equation in an intensive magnetic field

Vasily N. Rodionov

The modified Dirac-Pauli equations, which are introduced by means of \({\gamma_5}\)-mass factorization of the ordinary Klein-Gordon operator, are considered. We also take into account the interaction of fermions with the intensive homogenous magnetic field focusing attention to their (g-2) gyromagnetic factor. The basis of this approach is developing of methods for study of the structure of regions of unbroken \(\cal PT\) symmetry of Non-Hermitian Hamiltonians which be no studied earlier. For that, without the use of perturbation theory in the external field the exact energy spectra are deduced with regard to spin effects of fermions. We also investigate the unique possible of experimental observability the non-Hermitian restrictions in the spectrum of mass consistent with the conjecture Markov about Maximal Mass. This, in principal will may allow to find out the existence of an upper limit value in spectrum masses of elementary particles and confirm or deny the significance of the Planck mass.

High Energy Physics – Theory (hep-th); Quantum Physics (quant-ph)

Equivalence of the effective Hamiltonian approach and the Siegert boundary condition for resonant states

Naomichi Hatano

Two theoretical methods of finding resonant states in open quantum systems, namely the approach of the Siegert boundary condition and the Feshbach formalism, are reviewed and shown to be algebraically equivalent to each other for a simple model of the T-type quantum dot. It is stressed that the seemingly Hermitian Hamiltonian of an open quantum system is implicitly non-Hermitian outside the Hilbert space. The two theoretical approaches extract an explicitly non-Hermitian effective Hamiltonian in a contracted space out of the seemingly Hermitian (but implicitly non-Hermitian) full Hamiltonian in the infinite-dimensional state space of an open quantum system.

Quantum Physics (quant-ph); Mesoscale and Nanoscale Physics (cond-mat.mes-hall); Mathematical Physics (math-ph); Nuclear Theory (nucl-th)

Cavity controlled spectral singularity

K. Nireekshan Reddy, S. Dutta Gupta

We study theoretically a PT-symmetric saturable balanced gain-loss system in a ring cavity configuration. The saturable gain and loss are modeled by two-level medium with or without population inversion. We show that the specifics of the spectral singularity can be fully controlled by the cavity and the atomic detuning parameters. The theory is based on the mean-field approximation as in standard theory of optical bistability. Further, in the linear regime we demonstrate the regularization of the singularity in detuned systems, while larger input power levels are shown to be adequate to limit the infinite growth in absence of detuning

Optics (physics.optics)