Zero width resonance (spectral singularity) in a complex PT-symmetric potential

We show that the complex PT-symmetric potential V (x) = -V-1 sech(2)x + iV(2) sech x tanh x entails a single zero-width resonance (spectral singularity) when V-1 + vertical bar V-2 vertical bar = 4n(2) + 4n + 3/4 (n = 0, 1, 2, 3, ... , vertical bar V-2 vertical bar > V-1 + 1/4) and the positive resonant energy is given as E* = 1/4 [vertical bar V-2 vertical bar - (1/4 + V-1)].