Field-Modified Quantum Potentials from Tridiagonal Representations: Analytical Spectra and Galerkin Simulations
Abstract
We develop an effective radial framework for analyzing new field-modified potential function of a charged spinless particle in collinear electric and magnetic fields. Using the weak- magnetic field approximation, we neglect the diamagnetic term and other angular projections under an explicit smallness condition ≪ ΔE, where E is the electric field, and the replacement of the angular factor by C=cosθ in the spherical coordinates. The resulting equation is therefore a field-aligned radial model whose validity is limited to angularly localized states. Within this setting, we apply the indirect mode (or search mode) of Tridiagonal representation approach in the Laguerre and Jacobi bases. In the Laguerre basis, the tridiagonal constraint generates linear-plus-inverse and linear-plus-quadratic radial potentials, and the coefficient recurrences are connected with Meixner--Pollaczek and Meixner polynomial type structures, therefore yielding the analytic spectra and expansion wavefunctions. In the Jacobi basis, the mapping of the radial half-line to a finite interval produces a singular confining effective potential and a three-term recurrence that is treated through a truncated generalized eigenvalue problem. The formulation provides analytically and numerically tractable benchmark models for field-modified potential functions, while keeping the assumptions, parameter restrictions, endpoint behavior, and convergence tests explicit. In the Laguerre basis the resultant spectra are obtained in closed form and are independent of the orientation and of the electric field, which enter through the shape of the effective central potential, whereas in the Jacobi basis the spectrum is obtained numerically from a new three-term recurrence; in all cases the magnetic field enters through a linear Zeeman shift.
Related
- Semi-analytical representation of the Gaunt-modified Landau-Lifshitz equation in a plane wave
- Modified scattering for the 3D Hartree equation with singular external potentials
- Exact analytical spectra of an effective field-modified radial Schrödinger equation by the Nikiforov--Uvarov method
- Nonnegativity of the second largest eigenvalue of 4 × 4 tridiagonal stochastic matrices
- The topology of tridiagonal isospectral sets
BibTeX
@misc{osunmusanmi2026fieldmodified,
title = {{Field-Modified Quantum Potentials from Tridiagonal Representations: Analytical Spectra and Galerkin Simulations}},
author = {Tunde Joseph Osunmusanmi and Berihu Teklu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.23348},
eprint = {2609.23348},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23348},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Field-Modified Quantum Potentials from Tridiagonal Representations: Analytical Spectra and Galerkin Simulations
AU - Tunde Joseph Osunmusanmi
AU - Berihu Teklu
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2609.23348
UR - https://arxiv.org/abs/2609.23348
AB - We develop an effective radial framework for analyzing new field-modified potential function of a charged spinless particle in collinear electric and magnetic fields. Using the weak- magnetic field approximation, we neglect the diamagnetic term and other angular projections under an explicit smallness condition ≪ ΔE, where E is the electric field, and the replacement of the angular factor by C=cosθ in the spherical coordinates. The resulting equation is therefore a field-aligned radial model whose validity is limited to angularly localized states. Within this setting, we apply the indirect mode (or search mode) of Tridiagonal representation approach in the Laguerre and Jacobi bases. In the Laguerre basis, the tridiagonal constraint generates linear-plus-inverse and linear-plus-quadratic radial potentials, and the coefficient recurrences are connected with Meixner--Pollaczek and Meixner polynomial type structures, therefore yielding the analytic spectra and expansion wavefunctions. In the Jacobi basis, the mapping of the radial half-line to a finite interval produces a singular confining effective potential and a three-term recurrence that is treated through a truncated generalized eigenvalue problem. The formulation provides analytically and numerically tractable benchmark models for field-modified potential functions, while keeping the assumptions, parameter restrictions, endpoint behavior, and convergence tests explicit. In the Laguerre basis the resultant spectra are obtained in closed form and are independent of the orientation and of the electric field, which enter through the shape of the effective central potential, whereas in the Jacobi basis the spectrum is obtained numerically from a new three-term recurrence; in all cases the magnetic field enters through a linear Zeeman shift.
ER -