Pseudo-hermitian and PT-symmetric quantum systems with energy-dependent potentials: Bound-state solutions and energy spectra

We introduce generalized versions of complex Scarf and Morse-type potentials that contain energy-dependent parameters. PT-symmetry and pseudo-hermiticity of the associated quantum systems are discussed, and a modified orthogonality relation and pseudo-norm are constructed. We show that despite energy-dependence, our systems can admit real energy spectra and normalizable solutions of bound-state type.