Resonances for steplike potentials: forward and inverse results

We consider resonances associated to the operator $-\frac{d^2}{d{x}^2}+V(x)$, where $V(x)=V_{+}$ if $x>x_M$ and $V(x)=V_{-}$ if $x<-x_M$, with $V_{+}≠V_{-}$. We obtain asymptotics of the resonance-counting function in several regions. Moreover, we show that in several situations, the resonances, $V_{+}$, and $V_{-}$ determine $V$ uniquely up to translation.