One dimensional scattering from two-piece rising potentials: a new avenue of resonances

TL;DR

This paper investigates one-dimensional scattering from rising potentials, revealing that two-piece configurations can support resonances, unlike smooth single-piece potentials, with implications for understanding metastable states and resonance phenomena.

Contribution

It demonstrates that two-piece rising potentials can host resonances, providing new models for studying Gamow states in one-dimensional quantum systems.

Findings

Resonances occur in two-piece rising potentials but not in smooth single-piece ones.

Complex-energy poles are identified in the reflection amplitude for various rising potential profiles.

Peaks in Wigner's time-delay and spatial eigenfunction catastrophes confirm the presence of resonances.

Abstract

We study scattering from potentials that rise monotonically on one side; this is generally avoided. We report that resonant states are absent in such potentials when they are smooth and single-piece having less than three real turning points (like in the cases of Morse oscillator, exponential and linear potentials). But when these potentials are made two-piece, resonances can occur. We further show that rising potentials next to a well/step/barrier are rich models of multiple resonances (Gamow's decaying states) in one- dimension. We use linear, parabolic and exponential profiles as rising part and find complex-energy poles, ${\cal E}_n=E_n-i\Gamma_n/2$ $(\Gamma_n > 0)$, in the reflection amplitude (s-matrix). The appearance of peaks in Wigner's (reflection) time-delay at $E=\epsilon_n$ (close to $E_n$) and spatial catastrophe in the eigenfunction confirm the existence of resonances and meta-stable states in these systems. PACS Nos.: 03.65.-w, 03.65.Nk