Real discrete spectrum in the complex non-PT-symmetric Scarf II potential

• Real discrete spectrum in a complex non-PT-symmetric potential.
• Real discrete spectrum yet Hamiltonian apparently not pseudo-Hermitian.
• Non-Hermiticity without PT-symmetry and pseudo-Hermiticity.
• One more novel feature of versatile complex Scarf II potential.
• Negative-energy poles in scattering co-efficients of a complex potential.

Hitherto, it is well known that complex PT-symmetric Scarf II has real discrete spectrum in the parametric domain of unbroken PT-symmetry. We reveal new interesting complex, non-PT-symmetric parametric domains of this versatile potential, V(x)V(x), where the spectrum is again discrete and real. Showing that the Hamiltonian, p2/2μ+V(x)p2/2μ+V(x), in the new cases is pseudo-Hermitian could be challenging, if possible.