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

TL;DR

This work analyzes charged quantum scattering in a helically twisted geometry in the presence of an Aharonov–Bohm flux, where the twist generates a geometry-induced Coulomb-like interaction and the flux introduces a topological phase. By starting from the Schrödinger equation in the twisted metric with minimal AB coupling, the radial equation is mapped onto the known 2D Coulomb+AB problem, yielding an exact, closed-form partial-wave $S$-matrix: $S_m^{(\lambda)}=S_m^{\mathrm{(AB)}}\frac{\Gamma(|m-\lambda|+\tfrac{1}{2}+i\eta_m)}{\Gamma(|m-\lambda|+\tfrac{1}{2}-i\eta_m)}$ with $\eta_m=\beta_{\text{geom}}/\kappa=-\omega k (m-\lambda)/\kappa$. The geometry and flux together determine the phase shifts $\delta_m^{(\lambda)}=\delta_m^{\mathrm{(AB)}}+\delta_m^{\mathrm{(geom)}}$, the scattering amplitude, and the differential and total cross sections, while poles of $S_m^{(\lambda)}$ reproduce the bound-state spectrum via the same analytic structure. The AB shift $m\to m-\lambda$ accounts for the topological effect, whereas the geometry-induced Coulomb strength $\propto \omega k (m-\lambda)$ provides a channel-dependent long-range interaction, allowing flux-tunable control of interference patterns in scattering. This analytic framework unifies scattering and bound-state information and suggests extensions to spin, relativistic dynamics, and engineered helical media.

Abstract

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.

Figures (4)

• Figure 1: Phase shift $\delta^{(\lambda)}_{m}(\kappa)$ as a function of the radial wave number $\kappa$ for $\omega=0.5$ and $k=1.0$. Each curve corresponds to a different value of the Aharonov--Bohm flux parameter, $\lambda=0$ (blue), $\lambda=0.25$ (orange), and $\lambda=0.5$ (green). Panel (a) shows the $s$-wave channel $m=0$, while panel (b) refers to the first angular channel $m=1$.
• Figure 2: Phase shift $\delta^{(\lambda)}_{m}(\omega)$ as a function of the torsion parameter $\omega$ for fixed $\kappa=1.0$ and $k=1.0$. Each curve corresponds to a different value of the Aharonov--Bohm flux parameter, $\lambda=0$ (blue), $\lambda=0.25$ (orange), and $\lambda=0.5$ (green). Panel (a) shows the $s$-wave channel $m=0$, while panel (b) refers to the first angular channel $m=1$.
• Figure 3: (Color online) Differential cross section $d\sigma/d\theta$ as a function of the scattering angle $\theta$ for the helically twisted Coulomb--Aharonov--Bohm problem. The curves are evaluated for $\kappa = 1$, $\omega = 0.7$, $k = 1$, and three values of the Aharonov--Bohm parameter, $\lambda = 0.0$, $0.25$, and $0.5$, using the partial--wave expansion with angular momenta in the range $m = -50,\ldots,50$. The angular domain is restricted to $0.5 \le \theta \le \pi - 0.5$ in order to exclude the forward and backward singular directions.
• Figure 4: (Color online) Regularized total cross section $\sigma_{\mathrm{tot}}^{(\mathrm{reg})}$ as a function of the radial wave number $\kappa$ for the helically twisted Coulomb--Aharonov--Bohm problem. The curves are evaluated for $\omega = 0.5$, $k = 1$, and three values of the Aharonov--Bohm parameter, $\lambda = 0$ (blue), $\lambda = 0.25$ (orange), and $\lambda = 0.5$ (green), using a partial-wave cutoff $m_{\mathrm{max}} = 50$.