arXiv subjects

Anand Aruna Kumar

Publications and source records attributed to Anand Aruna Kumar.

6 recordsLinked to original sources

arXiv2026

Bloch states of an infinite alternating-charge lattice under a transverse electric field

We consider an infinite one dimensional array of alternating charges embedded in a two-dimensional configuration space and subjected to a weak static electric field transverse to the lattice axis. Expansion of the Coulomb interaction about a lattice site gives a transverse restoring stiffness $κ\propto ζ(3)/L^3$, defining the oscillator length $a_y$, while Bloch periodicity is retained along the lattice direction and the transverse motion is represented in a harmonic basis. The Coulomb matrix elements are governed by the dimensionless scale $η_{r-s}$, which couples the reciprocal-lattice transfer to the transverse oscillator length. A regularized one dimensional extension is also introduced through Coulomb regularization, yielding a finite diagonal offset and an exponentially decaying even-transfer coupling in reciprocal space. Small $η_{r-s}$ yields logarithmic, parity-dependent transverse-state coupling, while large $η_{r-s}$ approaches the one-dimensional Coulomb limit. In the finite-order spectrum, the latter approaches the Kronig Penney reference toward the band edge while retaining distinct Brillouin zone curvature. The formulation therefore identifies two characteristic quantities of the alternating lattice: the transverse restoring stiffness and the one-dimensional Madelung form, while retaining the two dimensional structure required to describe the response to a transverse electric field. Keywords: alternating charge lattice; Bloch states; Coulomb interaction; transverse electric field; regularization; reciprocal space coupling; Brillouin zone spectrum.

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arXiv2026

Canonical regularization of the stationary Coulomb problem and an Aufbau-like spectral ordering

The stationary hydrogen atom has Coulomb degeneracy across orbital levels, whereas the Aufbau/Madelung ordering is an empirical, many-electron rule established in atomic physics. We examine the hydrogen atom through a regularized de Broglie--Bohm representation, in which stationary amplitude current constraints generate separable Sturm--Liouville branches. In this formulation, the radial, orbital, and magnetic sectors acquire canonical Langer-like inverse square corrections. The modified boundary value problems allow analytical solutions and produce a hydrogen-like spectrum with regularized radial and angular indices. Consequently, radial Coulomb quantization acquires an orbital dependent shift, lifting the Coulomb degeneracy and producing a spectral ordering that follows the Aufbau/Madelung sequence. On this basis, we construct the ordering of the regularized de Broglie--Bohm states and show that the spectral structure retains the standard degenerate Rydberg sequence in the l=0 sector. The separated amplitudes are represented by generalized special function branches, including the associated Laguerre, Legendre, and Bessel functions with non-integral parameters arising from regularized separation. Therefore, the treatment is intended as an analytical examination of spectral ordering in a regularized one center Coulomb problem rather than as a replacement for the many electron atomic structure theory. Keywords: de Broglie--Bohm representation; Coulomb spectrum; canonical regularization; Langer correction; Sturm--Liouville equations; Aufbau principle; Madelung ordering; associated Legendre functions; associated Laguerre functions; Bessel functions.

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arXiv2026

Stationary Bohmian superposition under amplitude and phase modulation

In this work, we examine the problem of stationary superposition in the Bohmian amplitude phase formulation, where amplitude and phase obey coupled nonlinear equations and direct linear superposition is not generally preserved. Considering two near degenerate stationary branches, we derive a hierarchical reduction in which the mean amplitude satisfies an Ermakov Pinney equation, while the difference amplitude evolves through a forced Mathieu Hill type modulation induced by energy and stationary current differences. It is shown that energy coherence alone does not uniquely determine phase coherence, since independent stationary currents continue to enter both the modulation and phase difference equations. For weak amplitude modulation, a Wronskian based stationary branch obtained from an Ermakov Pinney solution admits a controlled amplitude phase construction, leading to an algebraic phase representation and a Jacobi Anger spectral expansion. As a result, a linear spectral structure emerges through Bessel weighted amplitude and phase modulation. Such a representation is naturally suited for modelling aperture geometries, as illustrated by rectangular and parabolic slit reductions exhibiting Fresnel type phase chirp and modulation driven sidebands. The present construction therefore provides an analytical route by which linear spectral superposition reemerges from nonlinear Bohmian amplitude phase dynamics. Keywords Bohmian mechanics; stationary superposition; amplitude and phase modulation; nonlinear superposition; Mathieu Hill equation; Fourier Bessel expansion.

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arXiv2026

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 current components. 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, thereby establishing a natural framework for the description of stationary Bohmian dynamics in the Landau problem.

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arXiv2026

Ermakov-Lewis Invariants in Stationary Bohm-Madelung Quantum Mechanics

The Ermakov Pinney equation and its associated invariant are shown to arise naturally in stationary quantum mechanics when the Schrodinger equation is expressed in Bohm Madelung form and the Hamiltonian is diagonal and separable. Under these conditions, the stationary continuity constraint induces a nonlinear amplitude equation of Ermakov Pinney type in each degree of freedom, revealing a hidden invariant structure that is independent of whether the evolution parameter is time or space. By reformulating the separated stationary equations in Sturm Liouville form and applying Liouville normalization, we demonstrate that the quantum potential is encoded as a curvature contribution of the self adjoint operator rather than appearing as an additional dynamical term. This correspondence preserves the standard probabilistic predictions of quantum mechanics while yielding exact stationary Bohmian amplitudes and their associated invariants. The resulting invariant-based formulation provides stationary guiding fields and clarifies the ontological status of Bohmian amplitudes as geometrically encoded structures rather than auxiliary dynamical additions. The results further show that stationary constrained Bohm Madelung systems naturally admit variational formulations whose extremals preserve the Ermakov Lewis invariant.

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arXiv2026

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 $μ$. 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 $p(x)x \to μ/2$ 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 $μ=\hbar$, 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.

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