  We point out that PT-symmetric potentials $V_{PT}(x)$ having imaginary
asymptotic saturation: $V_{PT}(x=\pm \infty) =\pm i V_1, V_1 \in \Re$ are
devoid of scattering states and spectral singularity. We show the existence of
real (positive and negative) discrete spectrum both with and without complex
conjugate pair(s) of eigenvalues (CCPEs). If the states are arranged in the
ascending order or real part of discrete eigenvalues, the initial states have
few nodes but latter ones oscillate fast. Both real and imaginary parts of
$\psi(x)$ vanish asymptotically, $|\psi(x)|$ for the CCPEs are asymmetric and
for real energies these are symmetric about origin. For CCPEs $E_{\pm}$ the
eigenstates $\psi_{\pm}$ follow an interesting property that $|\psi_+(x)|= N
|\psi_-(-x)|, N \in \Re^+$. We remark that, the fast oscillating real discrete
energy states discussed are likely to be confused with: reflectionless states,
one dimensional version of von Neumann states of Hermitian and spectral
singularity state of complex PT-symmetric potentials.
