Various scattering properties of a new PT-symmetric non-Hermitian potential

We complexify a 1-d potential V(x)=V₀cosh²μ{tanh(x−μd)/d]+tanh(μ)}²$V\left(x\right)=V_0{cosh}^2\mu {\{tanh(x-\mu d)/d]+tanh\left(\mu \right)\}}^2$ which exhibits bound, reflecting and free states to study various properties of a non-Hermitian system. This potential turns out a PT-symmetric non-Hermitian potential when one of the parameters (μ,d)$(\mu ,d)$ becomes imaginary. For the case of μ→iμ$\mu \to i\mu$, we have an entire real bound state spectrum. Explicit scattering states are constructed to show reciprocity at certain discrete values of energy even though the potential is not parity symmetric. Coexistence of deep energy minima of transmissivity with the multiple spectral singularities (MSS) is observed. We further show that this potential becomes invisible from the left (or right) at certain discrete energies. The penetrating states in the other case (d→id$d\to id$) are always reciprocal even though it is PT-invariant and no spectral singularity (SS) is present in this case. The presence of MSS and reflectionlessness is also discussed for the free states in the later case.