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## 1 Introduction 

Recent developments on non-Hermitian Hamiltonians have documented that Hermiticity is no more a necessary condition to secure the reality of the spectrum [1-43]. Such developments are very much inspired by the nowadays known as the Bender’s and Boettcher’s [1] conjecture in relaxing Hermiticity condition and introducing the concept of PT -symmetric quantum mechanics (PTQM). Where, P denotes space reflection: x −→ − x (i.e., parity operator) and T mimics the time-reversal: i −→ − i . More specifically, if ρ = PT and ρ H ρ − 1 = H , then H is PT -symmetric. Moreover, if ρ Ψ = ± Ψ (i.e., Ψ retains PT -symmetry) the eigenvalues of a PT -symmetric Hamiltonian are real, otherwise the eigenvalues come out in complex-conjugate pairs (a phenomenon known as spontaneous breakdown of PT -symmetry). 

Such a PTQM theory, nevertheless, has stimulated intensive research on the non-Hermitian Hamiltonians and led to the so-called pseudo-Hermitian Hamil- tonians (i.e., Hamiltonians satisfying ξ H ξ − 1 = H † or ξ H = H $^{†}$ξ , where ξ is a Hermitian invertible linear operator and ( $^{†}$) denotes the adjoint) by Mostafazadeh [20-25] which form a broader class of non-Hermitian Hamiltoni- ans with real spectra that encloses within those PT -symmetric ones. Moreover, not restricting ξ to be Hermitian (cf., e.g., Bagchi and Quesne [38]), and lin- ear and/or invertible (cf., e.g., Solombrino [32], Fityo [33], and Mustafa and Mazharimousavi [34-37]) would lead to real spectra. 

In the process, on the other hand, some quantum mechanical models of certain exceptional PT -symmetric complex interactions, i.e., a PT -symmetric potential satisfies 

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