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

Abstract

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 U0 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 τ(E) as a function of the peak energy E. We display the resonant behavior of τ (E) at energies close to U0. By analyzing the asymptotic behavior for large energies of the eigenfunctions of the continuous spectrum we also show that, as expected, τ(E) approaches the classical value for E → ∞, thus diverging for the step-linear case and vanishing for the step-exponential one

Scattering length and effective range of microscopic two-body potentials

“…( 13) can be written in terms of the spherical Bessel functions of the first and second kind, usually denoted by j l (x) and n l (x), respectively. The solution is of the form u l (r) = c l rj l (kr) + c l rn l (kr), (14) where c l and c l are constants. It is convenient to write the solution in terms of a linear combination of spherical Bessel, h…”

“…Detailed studies regarding analytical solutions in simple onedimensional (1D) potentials are discussed in Refs. [12][13][14]. For 1D potentials that do not support bound states, quantities such as the reflection and transmission coefficients are typically calculated.…”