2025/12/21 by Anand Aruna Kumar, Kumar, Anand Aruna, S. K. Srivatsa +3
Physics and Astronomy · #Quantum Mechanics and Applications #Quantum Mechanics and Non-Hermitian Physics #Statistical Mechanics and Entropy
paper · doi:10.48550/arxiv.2512.18555
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 μ. 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 → μ/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 μ=ℏ, 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.