Solvable models of an open well and a bottomless barrier: one-dimensional exponential potentials

We present one-dimensional potentials $V\left(x\right)=V_0(e^{2|x|/a}-1)$ as solvable models of a well $(V_0>0)$ and a barrier ($V_0<0$). Apart from being a new addition to solvable models, these models are instructive for finding bound and scattering states from the analytic solutions of the Schrödinger equation. The exact analytic (semi-classical and quantal) forms for bound states of the well and reflection/transmission $\left(R/T\right)$ coefficients for the barrier have been derived. Interestingly, the crossover energy E _c where $R\left(E_c\right)=1/2=T\left(E_c\right)$ may occur above, below or at the barrier-top. A connection between the poles of these coefficients and the bound-state eigenvalues of the well is also demonstrated. (paper)