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T. J. Taiwo

Publications and source records attributed to T. J. Taiwo.

10 recordsLinked to original sources
arXiv2025

Perturbative nonlinear J-matrix method of scattering in two dimensions

We introduce a perturbative formulation for a nonlinear extension of the J-matrix method of scattering in two dimensions. That is, we obtain the scattering matrix for the time-independent nonlinear Schrödinger equation in two dimensions with circular symmetry. The formulation relies on the linearization of products of orthogonal polynomials and on the utilization of the tools of the J-matrix method. Gauss quadrature integral approximation is instrumental in the numerical implementation of the approach. We present the theory for a general ψ^{2n + 1} nonlinearity, where n is a natural number, and obtain results for the cubic and quintic nonlinearities, ψ^3 and ψ^5. At certain value(s) of the energy, we observe the occurrence of bifurcation with two stable solutions. This curious and interesting phenomenon is a clear signature and manifestation of the underlying nonlinearity.

quant-ph↗
arXiv2025

Spontaneous symmetry and antisymmetry breaking in a ring with two potential barriers

We propose a fundamental setup for the realization of spontaneous symmetry breaking (SSB) and spontaneous antisymmetry breaking (SASB) in the framework of the nonlinear Schroedinger equation with the self-attractive and repulsive cubic term, respectively, on a one-dimensional ring split in two mutually symmetric boxes by delta-functional potential barriers, placed at opposite points. The system is relevant to optics and BEC. The spectrum of the linearized system is found in analytical and numerical forms. SSB and SASB are predicted by dint of the variational approximation, and studied in the numerical form. A particular stable solution, which demonstrates strong asymmetry, is found in an exact form. In the system with the attractive nonlinearity, the SSB of the symmetric ground state is initiated by the modulational instability. It creates stationary asymmetric states through a supercritical bifurcation. In the self-repulsive system, SASB makes the lowest antisymmetric excited state unstable, transforming it into an antisymmetry-breaking oscillatory mode.

nlin.PS↗
arXiv2022

Energy spectrum design and potential function engineering

Starting with an orthogonal polynomial sequence $\{p_n(s)\}_{n=0}^\infty$ that has a discrete spectrum, we design an energy spectrum formula, $E_k = f (s_k)$, where $|{s_k\}$ is the finite or infinite discrete spectrum of the polynomial. Using a recent approach for doing quantum mechanics based, not on potential functions but, on orthogonal energy polynomials, we give a local numerical realization of the potential function associated with the chosen energy spectrum. In this work, we select the three-parameter continuous dual Hahn polynomial as an example. Exact analytic expressions are given for the corresponding bound states energy spectrum, scattering states phase shift, and wavefunctions. However, the potential function is obtained only numerically for a given set of physical parameters.

quant-ph↗
arXiv2020

Confined systems associated with the discrete Meixner polynomials

Using a formulation of quantum mechanics based on orthogonal polynomials in the energy and physical parameters, we study quantum systems totally confined in space and associated with the discrete Meixner polynomials. We present several examples of such systems, derive their corresponding potential functions, and plot some of their bound states.

quant-ph↗
arXiv2017

The infinite square well in a reformulation of Quantum mechanics without potential function

Using a recent reformulation of quantum mechanics where the potential function is not required, we are able to obtain the energy spectrum and wave function associated with the infinite square well analytically. Therefore, this work constitutes an example of how to establish the correspondence between this approach and the standard formulation of quantum mechanics.

math-ph↗
arXiv2016

Asymptotic Limit of Continuous Dual Hahn Polynomial

Using Darboux method, we obtain an asymptotic limit for Continuous Dual Hahn Polynomial.The basic concept is to construct a comparison function which is the singular part of the generating function; then expand the comparison function. With little simplification and modification,the asymptotic limit is achieved.

math-ph↗
arXiv2016

The Wilson-Racah Quantum System

Using a recent formulation of quantum mechanics without potential function, we present a four-parameter system associated with the Wilson and Racah polynomials. The continuum scattering states are written in terms of the Wilson polynomials whose asymptotics gives the scattering amplitude and phase shift. On the other hand, the finite number of discrete bound states are associated with the Racah polynomials. We are honored to dedicate this work to Prof. Hashim A. Yamani on the occasion of his 70th birthday.

quant-ph↗