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

We show that the complex PT-Symmetric potential, $V(x)=-V_1 {sech}^2x + iV_2 {sech}x ~\tanh x,$, entails a single zero-width resonance (spectral singularity) when $V_1+|V_2|=4n^2+4n+{3\over 4}(n=1,2,3.., |V_2|>|V_1|+ {{sgn}(V_1) \over 4})$ and the positive resonant energy is given as $E_*={1 \over 4}[|V_2|-(1/4+V_1)]$.

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