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arXiv:2512.18555

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

Anand Aruna Kumar, S. K. Srivatsa, Rajesh Tengli

quant-phSubmitted Dec 21, 2025• Announced Dec 23, 2025• 0 citations• Revised Feb 22, 2026 (v2)
v1 · Dec 21, 2025v2 · Feb 22, 2026v3 · Mar 5, 2026
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Abstract

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 Schrodinger-type equation with a parametric information coupling mu. Beyond the density-based formulation, we introduce a reduced (shell) variational principle for the local momentum field, obtained by imposing stationary flux closure. This reduction isolates the regularisation mechanism at the level of spatial momentum flow and leads to constrained Euler Lagrange equations governing admissible amplitude configurations. The resulting first integral possesses an elliptic (Weierstrass) structure, whose admissible asymptotic branch enforces a universal canonical relation px=mu over 2 near amplitude zeros. This canonical product emerges dynamically from amplitude regularity and conserved flux, rather than being postulated. 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 which, under physical identification mu=hbar naturally reduces to the reduced Compton wavelength. Thus, canonical Bohmian regularisation is interpreted as a variational admissibility condition on density dynamics, providing structurally stable analytical branches and modified yet consistent energy spectra within stationary dBB mechanics.

Journal reference
International Journal of Quantum Foundations, Vol 12, Issue 2, Apr 2026
Comments
13 pages, 1 table, 1 figure, major revisions from first version

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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 Schrodinger-type equation with a parametric information coupling mu.
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