Non-catastrophic resonant states in one dimensional scattering from a rising exponential potential

TL;DR

This paper explores scattering from a novel two-piece rising exponential potential, revealing non-catastrophic resonant states with complex energy poles that produce observable peaks in reflection time-delay without spatial divergence.

Contribution

It introduces and analyzes a new class of rising exponential potentials, demonstrating the existence of non-catastrophic resonant states with complex energies and unique scattering properties.

Findings

Resonant states occur at complex energy poles of the reflection amplitude.

Reflection time-delay peaks at resonance energies, indicating resonances.

Eigenstates do not exhibit spatial catastrophe at resonance energies.

Abstract

Investigation of scattering from rising potentials has just begun, these unorthodox potentials have earlier gone unexplored. Here, we obtain reflection amplitude ($r(E)$) for scattering from a two-piece rising exponential potential: $V(x\le 0)=V_1[1-e^{-2x/a}], V(x > 0)=V_2[e^{2x/b}-1]$, where $V_{1,2}>0$. This potential is repulsive and rising for $x>0$; it is attractive and diverging (to $-\infty$) for $x<0$. The complex energy poles (${\cal E}_n= E_n-i\Gamma_n/2, \Gamma_n>0$) of $r(E)$ manifest as resonances. Wigner's reflection time-delay displays peaks at energies $E(\approx E_n$) but the eigenstates do not show spatial catastrophe for $E={\cal E}_n$.