Current conservation and amplitude regularisation of the Landau problem: Bohm--Madelung description

This work investigates the dynamics of a charged particle in a uniform magnetic field within the Bohm--Madelung formulation of quantum mechanics. In this representation, the stationary Schrodinger equation separates into coupled amplitude and phase equations, where the amplitude sector admits a Sturm--Liouville structure supporting Ermakov--Lewis invariants. The analysis considers two complementary regularisation schemes: a global Fisher--information--based regularisation and a local canonical (shell) Bohm regularisation derived from stationary flux closure. These are applied within distinct classes of stationary flow, characterised by vanishing and nonvanishing component currents. It is shown that the radial and axial sectors remain globally regularisable, preserving analytic structure across the domain. In contrast, the azimuthal sector develops a nonseparable, generally complex-valued amplitude structure due to gauge-induced coupling. Nevertheless, a consistent local regularity is recovered at the level of canonical branches, where amplitude--momentum relations organise the solution in a well-defined manner. Regularisation thus acts as a structural reorganisation mechanism in amplitude space, preserving the Landau spectral scale while reorganising the flux-sector structure through branch-wise amplitude--momentum relations, thus establishing branch-wise organisation as a natural framework for describing stationary Bohmian dynamics in the Landau problem.