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

We present one-dimensional potentials V(x)= V-0(e(2 vertical bar x vertical bar/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 Schrodinger equation. The exact analytic (semi-classical and quantal) forms for bound states of the well and reflection/transmission (R/T) coefficients for the barrier have been derived. Interestingly, the crossover energy E-c where R(E-c)= 1/2= T(E-c) 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.