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
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 ##IMG##
[http://ej.iop.org/images/0143-0807/36/4/048001/ejp512438ieqn1.gif]
{$({{mathcal{E}}_{n}}={{E}_{n}}-{rm i}{{Gamma }_{n}}/2)$} . These are also spatially catastrophic
states that manifest as peaks/wiggles in Wigner’s reflection time delay at ##IMG##
[http://ej.iop.org/images/0143-0807/36/4/048001/ejp512438ieqn2.gif] {$E={{epsilon }_{n}}approx
{{E}_{n}}$} . 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.