Scattering and delay time for 1D asymmetric potentials: The step-linear and the step-exponential case

Espalhamento e tempo de retardo para potenciais unidimensionais assimétricos: Casos linear e exponencial por partes

We analyze the quantum-mechanical behavior of a system described by a one-dimensional asymmetric potential constituted by a step plus (i) a linear barrier or (ii) an exponential barrier. We solve the energy eigenvalue equation by means of the integral representation method, classifying the independent solutions as equivalence classes of homotopic paths in the complex plane. We discuss the structure of the bound states as function of the height $U_{\text{0}}$ of the step and we study the propagation of a sharp-peaked wave packet reflected by the barrier. For both the linear and the exponential barrier we provide an explicit formula for the delay time $\tau \left(E\right)$ as a function of the peak energy E. We display the resonant behavior of $\tau \left(E\right)$ at energies close to $U_{\text{0}}$. By analyzing the asymptotic behavior for large energies of the eigenfunctions of the continuous spectrum we also show that, as expected, $\tau \left(E\right)$ approaches the classical value for $E\to \infty$, thus diverging for the step-linear case and vanishing for the step-exponential one.

Keywords:
integral transforms; special functions; solutions of wave equations: bound states; scattering theory