A regularisation method to obtain analytical solutions to the de Broglie Bohm wave equation

We develop a variational regularisation framework that enables analytical solutions of the stationary de~Broglie–Bohm wave equation. The formulation begins with a Fisher-information-augmented action functional for the probability density and phase fields, yielding the Madelung (Hamilton–Jacobi and continuity) equations and, upon complex recombination, a Schrödinger-type equation with a parametric information coupling $\mu$. Beyond this density-based formulation, we introduce a variational regularisation scheme for the de~Broglie–Bohm equations that combines a global Fisher-information regularisation at the level of the action functional with a shell-level regularisation arising from stationary flux closure. This reduction isolates the regularisation mechanism in the spatial momentum flow and yields constrained Euler–Lagrange equations governing admissible amplitude configurations. The resulting first integral possesses an elliptic structure whose admissible asymptotic branch enforces a universal canonical relation $p(x)x\to \mu \mathrm{/}2$ near amplitude zeros. The framework yields closed-form analytical solutions for standard potentials and reveals a systematic inverse-square regularising term in the effective potential. The associated elliptic discriminant defines a geometric length scale that, for $\mu =\mathrm{\hslash }$, naturally reduces to the reduced Compton wavelength. Canonical Bohmian regularisation therefore appears as a variational admissibility condition on density dynamics, producing structurally stable analytical branches and modified yet consistent energy spectra within stationary dBB mechanics.