Scattering in the PT-symmetric Coulomb potential

TL;DR

This paper investigates scattering phenomena in a PT-symmetric Coulomb potential along a complex trajectory, deriving analytic expressions for transmission and reflection, and analyzing bound states and their energies.

Contribution

It introduces a novel approach to scattering in PT-symmetric Coulomb potentials using a complex trajectory, providing explicit formulas for scattering coefficients and bound states.

Findings

Transmission and reflection coefficients are obtained analytically.

Scattering exhibits handedness effect similar to other PT-symmetric systems.

Bound-state energies are derived from transmission coefficient poles.

Abstract

Scattering on the ${\cal PT}$-symmetric Coulomb potential is studied along a U-shaped trajectory circumventing the origin in the complex $x$ plane from below. This trajectory reflects ${\cal PT}$ symmetry, sets the appropriate boundary conditions for bound states and also allows the restoration of the correct sign of the energy eigenvalues. Scattering states are composed from the two linearly independent solutions valid for non-integer values of the 2L parameter, which would correspond to the angular momentum in the usual Hermitian setting. Transmission and reflection coefficients are written in closed analytic form and it is shown that similarly to other ${\cal PT}$-symmetric scattering systems the latter exhibit handedness effect. Bound-state energies are recovered from the poles of the transmission coefficients.