Scattering of a relativistic scalar particle by a cusp potential

TL;DR

This paper analyzes how a relativistic scalar particle scatters off a cusp potential by solving the Klein-Gordon equation, revealing conditions for transmission resonances and their dependence on potential shape and phase shifts.

Contribution

It provides an exact solution for scattering in a cusp potential using Whittaker functions and derives new conditions for transmission resonances and zero-reflection states.

Findings

Transmission resonances depend on potential shape.

Zero-reflection conditions relate to phase shifts.

Resonances linked to half-bound states at low momentum.

Abstract

We solve the Klein-Gordon equation in the presence of a spatially one-dimensional cusp potential. The scattering solutions are obtained in terms of Whittaker functions and the condition for the existence of transmission resonances is derived. We show the dependence of the zero-reflection condition on the shape of the potential. In the low momentum limit, transmission resonances are associated with half-bound states. We express the condition for transmission resonances in terms of the phase shifts.