We investigate some questions on the construction of \(\eta\) operators for pseudo-Hermitian Hamiltonians. We give a sufficient condition which can be exploited to systematically generate a sequence of \(\eta\) operators starting from a known one, thereby proving the non-uniqueness of \(\eta\) for a particular pseudo-Hermitian Hamiltonian. We also study perturbed Hamiltonians for which \(\eta\)’s corresponding to the original Hamiltonian still work.

http://arxiv.org/abs/1401.5255
Quantum Physics (quant-ph)

Using canonical transformations we obtain localized (in space) exact solutions of the nonlinear Schrodinger equation (NLSE) with space and time modulated nonlinearity and in the presence of an external potential depending on space and time. In particular we obtain exact solutions of NLSE in the presence of a number of non Hermitian \(\mathcal{PT}\) symmetric external potentials.

http://arxiv.org/abs/1307.7591

Pattern Formation and Solitons (nlin.PS); Mathematical Physics (math-ph)

We report the existence and properties of localized modes described by nonlinear Schroedinger equation with complex PT-symmetric Rosen-Morse potential well. Exact analytical expressions of the localized modes are found in both one dimensional and two-dimensional geometry with self-focusing and self-defocusing Kerr nonlinearity. Linear stability analysis reveals that these localized modes are unstable for all real values of the potential parameters although corresponding linear Schroedinger eigenvalue problem possesses unbroken PT-symmetry. This result has been verified by the direct numerical simulation of the governing equation. The transverse power flow density associated with these localized modes has also been examined.

http://arxiv.org/abs/1304.2105
Quantum Physics (quant-ph); Pattern Formation and Solitons (nlin.PS); Optics (physics.optics)

We study generalized Dirac oscillators with complex interactions in \((1+1)\) dimensions. It is shown that for the choice of interactions considered here, the Dirac Hamiltonians are \(\eta\) pseudo Hermitian with respect to certain metric operators \(\eta\). Exact solutions of the generalized Dirac Oscillator for some choices of the interactions have also been obtained. It is also shown that generalized Dirac oscillators can be identified with Anti Jaynes Cummings type model and by spin flip it can also be identified with Jaynes Cummings type model.

http://arxiv.org/abs/1301.2035
Mathematical Physics (math-ph); High Energy Physics – Theory (hep-th); Quantum Physics (quant-ph)

Using an appropriate change of variable, the Schroedinger equation is transformed into a second-order differential equation satisfied by recently discovered Jacobi type \(X_m\) exceptional orthogonal polynomials. This facilitates the derivation of infinite families of exactly solvable Hermitian as well as non-Hermitian trigonometric and hyperbolic Scarf potentials whose bound state solutions are associated with the aforesaid exceptional orthogonal polynomials. These infinite families of potentials are shown to be extensions of the conventional trigonometric and hyperbolic Scarf potentials by the addition of some rational terms characterized by the presence of classical Jacobi polynomials. All the members of a particular family of these `rationally extended polynomial-dependent’ potentials have the same energy spectrum and possess translational shape invariant symmetry. The obtained non-Hermitian potentials are shown to be quasi-Hermitian in nature ensuring the reality of the associated energy spectra.

http://arxiv.org/abs/1210.0119
Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Using the method of point canonical transformation, we derive some exactly solvable rationally extended quantum Hamiltonians which are non-Hermitian in nature and whose bound state wave functions are associated with Laguerre and Jacobi-type \(X_1\) exceptional orthogonal polynomials. These Hamiltonians are shown, with the help of imaginary shift of co-ordinate: \(e^{-\alpha p} x e^{\alpha p} = x+ i \alpha\), to be both quasi and pseudo-Hermitian. It turns out that the corresponding energy spectra is entirely real.

http://arxiv.org/abs/1205.5860
Mathematical Physics (math-ph); Quantum Physics (quant-ph)

We show that the non Hermitian Black-Scholes Hamiltonian and its various generalizations are eta-pseudo Hermitian. The metric operator eta is explicitly constructed for this class of Hamitonians. It is also shown that the e?ective Black-Scholes Hamiltonian and its partner form a pseudo supersymmetric system.

http://arxiv.org/abs/1112.3217
General Finance (q-fin.GN)

We apply the factorization technique developed by Kuru and Negro [Ann. Phys. 323 (2008) 413] to study complex classical systems. As an illustration we apply the technique to study the classical analogue of the exactly solvable PT symmetric Scarf II model, which exhibits the interesting phenomenon of spontaneous breakdown of PT symmetry at some critical point. As the parameters are tuned such that energy switches from real to complex conjugate pairs, the corresponding classical trajectories display a distinct characteristic feature – the closed orbits become open ones.

http://arxiv.org/abs/1101.0909
Quantum Physics (quant-ph)
Physics Letters A : vol. 375 (2011) p 452-457