New shape resonances in one dimension

Hitherto, a finitely thick barrier next to a well or a rigid wall has been considered the potential of simplest shape giving rise to resonances (metastable states) in one dimension: $x\in (-\mathrm{\infty },\mathrm{\infty })$. In such a potential, there are three real turning points at an energy below the barrier. Resonances are Gamow’s (time-wise) decaying states with discrete complex energies $({\mathcal{E}}_n=E_n-i{\mathrm{\Gamma }}_n\mathrm{/}2)$. These are also spatially catastrophic states that manifest as peaks/wiggles in Wigner’s reflection time-delay at $E={ϵ}_n\approx E_n$. Here we explore potentials with simpler shapes giving rise to resonances - two-piece rising potentials having just one-turning point. We demonstrate our point by using rising exponential profile in various ways.