mathcal{PT}$ Symmetric Hamiltonian Model and Dirac Equation in 1+1 dimensions

JPA 46 015302 2013 In this article, we have introduced a $\mathcal{PT}$ symmetric non-Hermitian Hamiltonian model which is given as $\hat{\mathcal{H}}=\omega (\hat{b}^{\dag}\hat{b}+1/2)+ \alpha (\hat{b}^{2}-(\hat{b}^{\dag})^{2})$ where $\omega$ and $\alpha$ are real constants, $\hat{b}$ and $\hat{b^...

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Bibliographic Details
Main Author Yesiltas, O
Format Journal Article
LanguageEnglish
Published 02.01.2013
Subjects
Mathematics - Mathematical Physics
Physics - Mathematical Physics
Physics - Quantum Physics
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Summary:JPA 46 015302 2013 In this article, we have introduced a $\mathcal{PT}$ symmetric non-Hermitian Hamiltonian model which is given as $\hat{\mathcal{H}}=\omega (\hat{b}^{\dag}\hat{b}+1/2)+ \alpha (\hat{b}^{2}-(\hat{b}^{\dag})^{2})$ where $\omega$ and $\alpha$ are real constants, $\hat{b}$ and $\hat{b^{\dag}}$ 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.
DOI:10.48550/arxiv.1301.0205