Propagating-wave approximation in two-dimensional potential scattering

TL;DR

This paper introduces a nonperturbative approximation for 2D potential scattering that neglects evanescent waves, simplifies calculations, and is exact for certain complex potentials, with connections to the Born and semiclassical approximations.

Contribution

It presents a novel approximation scheme for 2D scattering that replaces the potential with an energy-dependent nonlocal form, simplifying analysis and linking to known approximations.

Findings

Reduces to the first Born approximation for weak potentials.

Becomes valid at high energies, similar to semiclassical approximation.

Identifies an infinite class of complex potentials where the scheme is exact.

Abstract

We introduce a nonperturbative approximation scheme for performing scattering calculations in two dimensions that involves neglecting the contribution of the evanescent waves to the scattering amplitude. This corresponds to replacing the interaction potential $v$ with an associated energy-dependent nonlocal potential ${\mathscr{V}}_k$ that does not couple to the evanescent waves. The scattering solutions $\psi(\mathbf{r})$ of the Schr\"odinger equation, $(-\nabla^2+{\mathscr{V}}_k)\psi(\mathbf{r})=k^2\psi(\mathbf{r})$, has the remarkable property that their Fourier transform $\tilde\psi(\mathbf{p})$ vanishes unless $\mathbf{p}$ corresponds to the momentum of a classical particle whose magnitude equals $k$. We construct a transfer matrix for this class of nonlocal potentials and explore its representation in terms of the evolution operator for an effective non-unitary quantum system. We show that the above approximation reduces to the first Born approximation for weak potentials, and similarly to the semiclassical approximation, becomes valid at high energies. Furthermore, we identify an infinite class of complex potentials for which this approximation scheme is exact. We also discuss the appealing practical and mathematical aspects of this scheme.