Numerical study of non-relativistic quantum systems and small oscillations induced in a helically twisted geometry

TL;DR

This paper numerically investigates bound states of a non-relativistic particle in a helically twisted geometry, revealing how geometry and magnetic flux influence quantum spectra and confinement.

Contribution

It introduces a numerical approach to analyze quantum states in curved, torsion-induced geometries with magnetic flux, exploring various interaction scenarios.

Findings

Geometry alone can induce effective confinement.

Spectral properties depend on torsion and magnetic flux.

Systematic trends of energy levels are identified.

Abstract

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 Schr\"odinger 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.