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

TL;DR

This paper revisits the PT-symmetric complexified Scarf II potential, exploring spectral singularities, their properties, and constructing rationally-extended supersymmetric partners, revealing that SUSY preserves spectral singularities across different partner potentials.

Contribution

It provides new insights into spectral singularities of the PT-symmetric Scarf II potential and constructs novel rationally-extended SUSY partners, analyzing their spectral properties.

Findings

Identification of spectral singularities as zero-width resonances.

Construction of four rationally-extended SUSY partners.

Spectral singularities are preserved under SUSY transformations.

Abstract

The $\cal PT$-symmetric complexified Scarf II potential $V(x)= - V_1 \sech^{2}x + {\rm i} V_2 \sech x \tanh x$, $V_1>0$ , $V_{2}\neq 0$ is revisited to study the interplay among its coupling parameters. The existence of an isolated real and positive energy level that has been recently 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 $\cal PT$-symmetric or complex non-$\cal PT$-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.