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

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 $E={\cal E}_n$.