Geometry--gauge controlled dynamics and localization of a two-particle system on a helicoidal manifold

Abstract

We study the dynamics and localization of two oppositely charged particles constrained to a helicoidal manifold in a uniform magnetic field. The embedding-induced metric and the pullback of the ambient electromagnetic gauge potential, together with restriction to the reflection-symmetric longitudinal rest frame, lead to an exact reduction of the relative dynamics to a one-dimensional Hamiltonian with a coordinate-dependent kinetic term and a gauge-shifted momentum. We analyze the corresponding effective potential, turning points, and classically allowed regions, and determine how the geometric and magnetic parameters modify the bounded relative motion. For a regularized attractive interaction, we derive the zero-energy condition for localization around the symmetric configuration and obtain the local stiffness governing its stability. When this stiffness changes sign while the quartic coefficient remains positive, the symmetric minimum undergoes a pitchfork-type bifurcation to two symmetry-related finite-separation minima of the relative coordinate. Canonical quantization of the reduced Hamiltonian then gives the low-energy spectrum in the harmonic approximation and the associated zero-energy localization thresholds. At the critical stiffness, the quadratic term vanishes and the quartic term provides the leading contribution to the local low-energy scaling. These results establish how helicoidal geometry and magnetic coupling modify the relative localization and low-energy spectral properties of the two-particle system.