## $$\mathcal{PT}$$-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.

http://arxiv.org/abs/1406.3298

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.

http://arxiv.org/abs/1406.3179
Mathematical Physics (math-ph)

## PT-Symmetric Hamiltonian Model and Dirac Equation in 1+1 dimensions

O. Yesiltas

In this article, we have introduced a $$\mathcal{PT}$$-symmetric non-Hermitian Hamiltonian model which is given as $$\hat{\mathcal{H}}=\omega (\hat{b}^{\dagger}\hat{b}+1/2)+ \alpha (\hat{b}^{2}-(\hat{b}^{\dagger})^{2})$$ where $$\omega$$ and $$\alpha$$ are real constants, $$\hat{b}$$ and $$\hat{b^{\dagger}}$$ are first order differential operators. The Hermitian form of the Hamiltonian $$\mathcal{\hat{H}}$$ is obtained by suitable mappings and it is interrelated to the time independent one dimensional Dirac equation in the presence of position dependent mass. Then, Dirac equation is reduced to a Schr\”{o}dinger-like equation and two new complex non-$$\mathcal{PT}$$-symmetric vector potentials are generated. We have obtained real spectrum for these new complex vector potentials using shape invariance method. We have searched the real energy values using numerical methods for the specific values of the parameters.

http://arxiv.org/abs/1301.0205
Mathematical Physics (math-ph); Quantum Physics (quant-ph)