$\mathcal{PT}$ Symmetric Hamiltonian Model and Exactly Solvable Potentials

Abstract

Searching for non-Hermitian (parity-time)$\mathcal{PT}$-symmetric Hamiltonians \cite{bender} 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 \cite{susy1}. Mapping function $\rho$ is obtained for each potential case.