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arXiv:2602.05229

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

Augusto Tadeu P. de Araújo, Edilberto O. Silva

quant-phSubmitted Feb 5, 2026• Announced Feb 6, 2026
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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.

Comments
21 pages, 4 figures

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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 2D2D2D Coulomb +++ AB problem, with a geometry-induced Coulomb strength and the azimuthal quantum number shifted as m→m−λm\to m-λm→m−λ. We construct the exact scattering solutions, obtain closed expressions for the partial-wave SSS 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 SSS matrix is consistent with the bound-state quantization previously obtained for the helically twisted Coulomb-like problem.
We investigate bound states of a non-relativistic scalar particle in a three-dimensional helically twisted (torsional) geometry, considering both the free case and the presence of external radial interactions. The dynamics is described by the Schrödinger equation on a curved spatial background and, when included, by minimal coupling to a magnetic vector potential incorporating an Aharonov--Bohm flux. After separation of variables, the problem reduces to a one-dimensional radial eigenvalue equation governed by an effective potential that combines torsion-induced Coulomb-like and centrifugal-like structures with magnetic/flux-dependent terms and optional model interactions. Because closed-form analytic solutions are not reliable over the parameter ranges required for systematic scans, we compute spectra and eigenfunctions numerically by formulating the radial equation as a self-adjoint Sturm--Liouville problem and solving it with a finite-difference discretization on a truncated radial domain, with explicit convergence control. We analyze four representative scenarios: (i) no external potential, (ii) Cornell-type confinement, (iii) Kratzer-type interaction, and (iv) the small-oscillation regime around the minimum of a Morse potential. We present systematic trends of the low-lying levels as functions of the torsion parameter, magnetic field, and azimuthal sector, and we show that geometric couplings alone can produce effective confinement even in the absence of an external interaction.
Exact analytic solutions are found to the Dirac equation in 2+12+12+1 dimensions for a combination of an Aharonov-Bohm potential and the Lorentz three-vector and scalar Coulomb potentials. By means of the solutions obtained the relativistic quantum Aharonov-Bohm effect is studied for the free (in the presence of a Lorentz three-vector Coulomb potential) and bound fermion states. We obtain the total scattering amplitude in a combination of the Aharonov-Bohm and Lorentz three-vector Coulomb potentials as a sum of two scattering amplitudes. This modifies the expression for the standard Aharonov-Bohm cross section due to the interference of these two amplitudes with each other. We discuss that the observable quantities can be the phases of electron wave functions or the energies of bound states.
In the Aharonov–Casher (AC) scattering, a neutral particle interacts with an infinitesimally thin, long charge filament resulting in a phase shift. In the original AC treatment, a ∇ · E term proportional to the charge density at the filament's position is dropped from the Hamiltonian on the basis that the particle is banned from the filament, thus, the resulting Hamiltonian compares with the Aharonov–Bohm Hamiltonian of a scalar particle. Here, we consider AC scattering with this term included. Starting from the three-dimensional nonrelativistic Aharonov–Casher (AC) Schrödinger equation with the ∇ · E term included, we find the wave functions — in particular their singular component — the phase shifts and thus compute the scattering amplitudes and cross-sections. We show that singular solutions in the AC case appear only when the delta function interaction introduced is attractive regardless of the spin orientation of the particle. We find that the inclusion of this term does not introduce a structural difference in the general form of the cross-section even for polarized particles. Its mere effect, is in shifting the parameter N (the greatest integer in α) that appears in the cross-section, in the attractive case, by one. Interesting situation appears when N = 0, thus α=δ, in the case α≻0, and N = -1, so α = 1-δ in the case α≺0: At these values of the parameter N, where αis just any fraction, the cross-section for a particle polarized in the scattering plane to scatter in a state with the same polarization, is isotropic. It also vanishes, at these values of N, for transitions between same-helicity eigenstates. For these values of the parameter N and at the special values α = ±1/2, the cross-sections for both signs of α coincide. The main differences between this model and the "mathematically equivalent" spin-1/2 AB theory are outlined.
The exact wave functions that describe scattering of a charged particle by a confined magnetic field (Aharonov-Bohm effect) and by a Coulomb field are analyzed. It is well known that the usual procedure of finding asymptotic forms of these functions which admit a separation into a superposition of an incident plane wave and a circular or spherical scattered wave is problematic near the forward direction. It thus appears to be impossible to express the conservation of probability by means of an optical theorem of the usual kind. Both the total cross section and the forward scattering amplitude appear to be infinite. To address these difficulties we find a new representation for the asymptotic form of the Aharonov-Bohm wave function that is valid for all angles. Rather than try to define a cross section at forward angles, however, we work instead with the probability current and find that it is quite well behaved. The same is true for Coulomb scattering. We trace the usual difficulties to a nonuniformity of limits.
The exact energy spectrum and wave function of a charged particle moving in the Coulomb field and Aharonov-Bohm's magnetic flux are solved by the nonintegrable phase factor. The universal formula for the matrix elements of the radial operator ${r}^{\ensuremath{\alpha}}$ of arbitrary power $\ensuremath{\alpha}$ is given by an analytical solution. The difference between the classical limit of matrix elements of inverse radius in quantum mechanics and the Fourier components of the corresponding quantity for the pure Coulomb system in classical mechanics is examined in reference to the correspondence principle of Heisenberg. Explicit calculation shows that the influence of nonlocal Aharonov-Bohm effect exists even in the classical limit. The semiclassical quantization rule for systems containing the topological effect is presented in the light of Heisenberg's corresponding principle.