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.
In recent decades, the study of topological defects has attracted significant interest across several areas of physics and mathematics [1]. Besides their role as solutions of nonlinear differential equations, such defects may act as domain walls delimiting distinct regions, often with observable consequences. Representative examples include kinks, vortices, and skyrmion-like configurations [2][3][4][5]. In gravitation and cosmology, cosmic strings arise in the Abelian Higgs model and share structural similarities with vortex solutions in flat spacetime, connecting spontaneous symmetry breaking to macroscopic phenomena [1,3,5].
The investigation of quantum dynamics in nontrivial backgrounds has produced a large literature, including relativistic and non-relativistic hydrogen-like problems in spacetimes containing topological defects such as cosmic strings and global monopoles [6,7]. Several extensions address magnetic fields, Aharonov-Bohm-type effects, extra dimensions, position-dependent masses, and related spectral problems . In these contexts, curvature and torsion can induce effective interactions and modify quantum phases. In particular, torsion is naturally described in Einstein-Cartan-type geometries and also appears in condensed-matter analogs such as defects in elastic or crystalline media [31][32][33].
A twisted helical geometry has been proposed as a tractable model of a torsion-like defect and explored for bound states, numerical classification of levels, and related physical applications [34][35][36]. The present work focuses on a consistent numerical treatment of the bound-state problem in this background. Instead of relying on series truncations and special-function representations, we formulate the reduced radial equation as a self-adjoint Sturm-Liouville eigenvalue problem and compute the spectrum using finite-difference discretization with explicit convergence checks. This strategy is robust across parameter regimes and directly supports scans required for spectral plots.
The paper is organized as follows. In Section II we introduce the helically twisted metric. In Section III we derive the reduced radial equation from the curved-space Schrödinger equation with minimal coupling to a magnetic vector potential and Aharonov-Bohm flux, and we present the numerical eigenvalue formulation and discretization scheme used throughout. Sections IV-VI apply the same numerical machinery to external Cornell and Kratzer interactions and to the small-oscillation (quadratic-plus-linear) approximation of the Morse potential, respectively. Section VII summarizes the main numerical findings and physical trends.
We consider an axially symmetric three-dimensional geometry characterized by a torsion-like helical twist that couples the angular coordinate ϕ to the longitudinal coordinate z. The spatial line element is
where ω is a dimensionless parameter controlling the strength of the helical twist. The coordinates are 0 < r < ∞, 0 < ϕ ≤ 2π, and -∞ < z < ∞. The metric tensor and its inverse are
with determinant g = det(g ij ) = r 2 . Further geometric and physical properties of this background can be found in Refs. [34][35][36].
A. Curved-space Schrödinger equation with minimal coupling
We start from the stationary Schrödinger equation for a scalar particle of effective mass µ in the curved spatial background,
where g = det(g ij ) and D i = ∂ i -ieA i implements minimal coupling to the electromagnetic potential. We adopt a uniform magnetic field along z and include an Aharonov-Bohm flux through
with constant B 0 and flux Φ B [22,37,38].
Using the ansatz ψ(r, ϕ, z) = e imϕ e ikz ξ(r),
with m ∈ Z and k ∈ R, Eq. ( 3) reduces to
where
Equation (7) represents a potential energy term, which carries the geometric and electromagnetic information of the system. It is important to note that, if we set e → 0, the potential energy
, that is, a potential energy of a purely geometric nature. This characteristic is provided by the presence of the parameter ω in the potential energy. Furthermore, we can note that this interaction contains a Coulomb-type potential plus a centrifugal-type potential term, thus forming a geometric Kratzer-Fues-type potential, V KF , defined in form
In this particular case, taking k = 0, we can note that the potential V 1 is reduced even further, becoming a centrifugal-type potential and, consequently, the geometric effects are inhibited; the quantum particle is not influenced by the torsion of spacetime.
It is convenient to eliminate the first-derivative term by ξ(r) = f (r)/ √ r. Then Eq. ( 6) becomes a one-dimensional Schrödinger-like equation,
with
This redefinition is advantageous because it casts the radial problem into a standard self-adjoint one-dimensional form, where the spectral parameter E enters linearly and all geometric and electromagnetic effects are encoded in a single effective potential. The extra contributionℏ 2 /(8µr 2 ) originates solely from the removal of th
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