\(\mathcal{PT}\)-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)

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