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

TL;DR

This paper investigates quantum scattering and delay times for asymmetric 1D potentials combining a step with either a linear or exponential barrier, providing explicit formulas and analyzing resonant and asymptotic behaviors.

Contribution

It introduces an integral representation method to solve the eigenvalue problem and derives explicit delay time formulas for both potential types, highlighting their resonant and asymptotic properties.

Findings

Explicit delay time formulas for step-linear and step-exponential barriers.

Resonant behavior of delay time near the step height U_0.

Asymptotic delay time approaches classical limits at high energies.

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 U_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(E) as a function of the peak energy E. We display the resonant behavior of \tau(E) at energies close to U_0. By analyzing the asymptotic behavior for large energies of the eigenfunctions of the continuous spectrum we also show that, as expected, \tau(E) approaches the classical value for E -> \infty, thus diverging for the step-linear case and vanishing for the step-exponential one.