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