Comparative analysis of real and $\mathcal{P}\mathcal{T}$-symmetric Scarf potentials

Abstract

The Scarf I and Scarf II potentials are discussed within a common mathematical framework, which is then specified to handle the two potentials separately both in the conventional Hermitian and in the $\mathcal{P}\mathcal{T}$-symmetric setting. The physically admissible solutions are identified in each case together with the corresponding energy eigenvalues. Several main differences between the $\mathcal{P}\mathcal{T}$-symmetric Scarf I and II potentials are pointed out. These include the presence and absence of the quasi-parity quantum number, the sign of the pseudo-norm, the mechanism of the spontaneous breakdown of $\mathcal{P}\mathcal{T}$ symmetry and the non-$\mathcal{P}\mathcal{T}$ orthogonality of otherwise admissible solutions in the Scarf I potential. Similarities and differences with respect to the corresponding Hermitian systems are also pointed out.