Propagating-wave approximation in two-dimensional potential scattering

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 ${\mathcal{V}}_k$ that does not couple to the evanescent waves. The scattering solutions $\psi (\mathbf{r})$ of the Schrödinger equation, $(-{\nabla }^2+{\mathcal{V}}_k)\psi (\mathbf{r})=k^2\psi (\mathbf{r})$, 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.

arXiv comment18 pages, 1 figure, slightly expanded version, accepted for publication in Phys. Rev. A

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Cited by in corpus (2)

• Fundamental transfer matrix for electromagnetic waves, scattering by a planar collection of point scatterers, and anti-PT-symmetry
• Existence of the transfer matrix for a class of nonlocal potentials in two dimensions