An update on the PT-symmetric complexified Scarf II potential, spectral singularities and some remarks on the rationally extended supersymmetric partners

The PT-symmetric complexified Scarf II potential V (x) = -V(1)sech(2)x + iV(2)sechx tanh x, V-1 > 0, V-2 not equal 0, is revisited to study the interplay among its coupling parameters. The existence of an isolated real and positive energy level that has recently been identified as a spectral singularity or zero-width resonance is here demonstrated through the behaviour of the corresponding wavefunctions and some property of the associated pseudo-norms is pointed out. We also construct four different rationally extended supersymmetric partners to V (x), which are PT-symmetric or complex non-PT-symmetric according to the coupling parameters range. A detailed study of one of these partners reveals that SUSY preserves the V (x) spectral singularity existence.