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: $x \in(-\infty, \infty)$. 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 $({\cal E}_n = E_n -i\Gamma_n/2)$. These are also spatially catastrophic states that manifest as peaks/wiggles in Wigner's reflection time-delay at $E = \epsilon_n \approx E_n$. 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.