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

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