Quantum scattering in helically twisted geometries: Coulomb-like interaction and Aharonov-Bohm effect

We investigate the scattering of a charged quantum particle in a helically twisted background that induces an effective Coulomb-like interaction, in the presence of an Aharonov-Bohm (AB) flux. Starting from the nonrelativistic Schrödinger equation in the twisted metric, we derive the radial equation and show that, after including the AB potential, it can be mapped onto the same Kummer-type differential equation that governs the planar $2D$ Coulomb $+$ AB problem, with a geometry-induced Coulomb strength and the azimuthal quantum number shifted as $m\to m-λ$. We construct the exact scattering solutions, obtain closed expressions for the partial-wave $S$ matrix and phase shifts, and derive the corresponding scattering amplitude, differential cross section, and total cross section. We also show that the pole structure of the $S$ matrix is consistent with the bound-state quantization previously obtained for the helically twisted Coulomb-like problem.

💡 Research Summary

In this work the authors present a thorough analytical study of the scattering of a charged non‑relativistic quantum particle moving in a three‑dimensional helically twisted space while being threaded by an Aharonov–Bohm (AB) magnetic flux. The geometry is described by the metric
( ds^{2}=dr^{2}+r^{2}d\phi^{2}+(dz+\omega r d\phi)^{2}),
where the dimensionless torsion‑like parameter (\omega) mixes the azimuthal and longitudinal coordinates. By inserting the metric into the Laplace–Beltrami operator and minimally coupling the AB vector potential (A_{\phi}= \Phi/(2\pi r)) (with the dimensionless flux (\lambda = q\Phi/2\pi\hbar)), the Schrödinger equation reduces, after separation of variables (\Psi(r,\phi,z)=\frac{1}{\sqrt{2\pi L}}e^{im\phi}e^{ikz}\psi_{m}(r)), to a radial equation

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