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

TL;DR

This paper demonstrates that a specific complex PT-symmetric potential exhibits a zero-width resonance, known as a spectral singularity, under certain parameter conditions, with explicit formulas for the resonance energy.

Contribution

It identifies the conditions for spectral singularities in a complex PT-symmetric potential and derives explicit expressions for the resonant energy levels.

Findings

Spectral singularity occurs when V_1 + |V_2| = 4n^2 + 4n + 3/4.

Resonant energy is given by E_* = (|V_2| - (1/4 + V_1)) / 4.

Zero-width resonance appears at specific parameter values in the potential.

Abstract

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 $E_*={1 \over 4}[|V_2|-(1/4+V_1)]$.