PT-symmetric potentials with imaginary asymptotic saturation

We point out that PT-symmetric potentials $$V_{\mathrm {PT}}(x)$$ having imaginary asymptotic saturation, {PT}}(x=\pm \infty ) =\pm i V_1, V_1 \in {\mathbb {R}}$$ are devoid of scattering states and spectral singularity. show the existence real (positive negative) discrete spectrum both with without complex conjugate pair(s) eigenvalues (CCPEs). If eigenstates arranged in ascending order part eigenvalues, initial have few nodes but latter ones oscillate fast. Both parts $$\psi _n(x)$$ vanish asymptotically, $$|\psi _n(x)|$$ nodeless. For CCPEs, these asymmetric peaking on left (right) for energies symmetric at origin. CCPEs $$E_{\pm }$$ , _{\pm follow interesting property, _+(x)|= N |\psi _-(-x)|, {R}}^+$$ .