Lévy-index control of spectral singularities and coherent perfect absorption in non-Hermitian space-fractional quantum mechanics

Abstract

We investigate the scattering features of a non-Hermitian rectangular potential within the framework of space-fractional quantum mechanics. Using the Riesz fractional derivative, we analytically derive locus equations for spectral singularities (SSs) and their time-reversed counterparts, coherent perfect absorption (CPA), in a dimensionless complex-potential parameter space. This geometric locus formulation provides a transparent representation of the SS and CPA conditions and enables direct visualization of how fractional quantum dynamics modifies non-Hermitian scattering. We show that reducing the Lévy index $α$, which enhances nonlocal transport associated with Lévy-flight dynamics, systematically lowers the gain-loss strength required for the emergence of SSs and CPAs, while increasing the mode index further suppresses this threshold. In addition, for fixed potential parameters, we demonstrate that decreasing $α$ induces a blue shift of the SS energy, in direct agreement with earlier studies. From this perspective, the Lévy index $α$ emerges as a tunable control knob for SS-CPA settings in fractional non-Hermitian quantum systems. Beyond its quantum-mechanical setting, this study may find applications in fractional waveguides and metamaterials governed by fractional wave equations. This work also bridges the gap between non-Hermitian quantum mechanics and space-fractional quantum mechanics.

Figures (2)

• Figure 1: (Color online). Graph of the curve $\widetilde{\mathbbmss{S}}_{n}^{-}(\rho,\sigma,\alpha)$ in the $\rho-\sigma$ plane for different $n$ with the Lévy index (a) $\alpha=2$ and (b) $\alpha=1.000005$ ($\alpha \rightarrow 1$). The result shown in (a) is fully consistent with the findings reported in Ref. mostafazadeh2009resonance. It is evident from the plots that no spectral singularity exists for $\sigma < 0$, as expected. Although for both cases, all the curves merge at $\rho=1$ for $\sigma=0$, this point does not correspond to a spectral singularity, since the potential becomes real when $\sigma=0$. $\widetilde{\mathbbmss{S}}_{1}^{-}(\rho,\sigma,\alpha)$ has a vertical asymptote at $\rho=0.667$ in figure (a).
• Figure 2: (Color online). Illustration of the loci of the SSs defined by $\widetilde{\mathbbmss{S}}_{n}^{-}(\rho,\sigma,\alpha)$ in the $\rho-\sigma$ plane for (a) $n=2$ and (b) $n=3$, shown for different values of the Lévy index $\alpha$. Panels (c) and (d) display the corresponding loci of CPA, the time-reversed counterpart of SS, defined by $\widetilde{\mathbbmss{C}}_{n}^{-}(\rho,\sigma,\alpha)$ for $n=2$ and $n=3$, respectively, for the same set of $\alpha$ values. As expected from time-reversal symmetry, the SS loci are confined to the region $\sigma>0$, while the CPA loci occur for $\sigma<0$, indicating that SSs and CPA solutions arise for opposite signs of the non-Hermitian coupling.