Date of Defense
2-4-2026 11:00 AM
Location
Microsoft Teams
Document Type
Dissertation Defense
Degree Name
Doctor of Philosophy in Mathematics
College
College of Science
Department
Mathematical Sciences
First Advisor
Prof. Qasem Al-Mdallal
Keywords
Special functions and orthogonal polynomials, Tridiagonal representation approach, J matrix methods of scattering, linearization of orthogonal polynomials, basis elements, nonlinear differential equation, nonlinear Schrödinger equations.
Abstract
This dissertation investigates the application of the Tridiagonal Representation Approach (TRA) to linear systems and introduces, for the first time, an extension of the J-matrix method of scattering to nonlinear phenomena. The TRA provides an algebraic framework for solving second-order linear ordinary differential equations by combining algebraic structure with the analytical properties of orthogonal polynomials and special functions. Computationally, the method benefits from efficient numerical techniques for tridiagonal matrices. Within the TRA, solutions of differential equations, including the Schrödinger equation, are expressed as convergent expansions in square-integrable basis functions chosen to yield a tridiagonal representation of the differential operator. This reduces the problem to a three-term recursion relation for the expansion coefficients, solvable in terms of orthogonal polynomials. For physical systems, key observables such as energy spectra and scattering quantities follow directly from the properties of these polynomials.
The dissertation also examines the J-matrix theory of scattering, where the reference Hamiltonian is treated analytically using a tridiagonal basis representation. Assuming a nonsingular, short-range potential allows truncation within a finite basis subset, with accuracy controlled by basis parameters and size. Unlike variational approaches, the J-matrix method treats the reference Hamiltonian exactly while approximating only the interaction. The J-matrix method has been widely applied in scattering problems in atomic, molecular, and nuclear physics. This work extends its scope to nonlinear phenomena alongside the TRA
analysis of linear systems.
Progressively, we used the TRA to obtain new confined linear systems with a linear energy spectrum. We provided multiple 1D examples and one 3D example to demonstrate the process. We plotted a few wavefunctions of the lowest energy bound states in addition to providing a graphical depiction of the potential function for each confined system. In chapter 4, we treat the inverse problem where for a given choice of the energy spectrum formula E (z2) and a particular collection of basis functions in configuration space { ϕn |(𝑥)}, we provide a general technique for constructing a class of quantum systems associated with the three-parameter continuous dual Hahn orthogonal polynomials Snμ (z2; a, b) . The energy spectrum, the scattering phase shift, and the wavefunction are obtained analytically. However, the corresponding potential function can only be derived numerically and locally for a given set of physical parameters. By choosing the energy as a special function of the polynomial argument E (z2) in chapter 5 and 6 and for the first time in the published literature, we introduce a nonlinear extension of the J-matrix method of scattering. This formulation relies heavily on the linearization formulas for products of orthogonal polynomials. The nonlinear J matrix method (semi analytic) was presented to handle nonlinear Schrodinger equation using a simple toy model in chapter 5 whereas in chapter 6, a perturbative formulation of nonlinear J matrix method in 2D for the NLSE. All results obtained in all cases are new and interesting and they have been published. In the future, as a further research plan, we intend to develop the nonlinear tridiagonal representation approach and the time dependent tridiagonal representation approach.
Included in
APPLICATION OF THE TRIDIAGONAL REPRESENTATION APPROACH AND THE J MATRIX METHOD OF SCATTERING IN THEORETICAL PHYSICS
Microsoft Teams
This dissertation investigates the application of the Tridiagonal Representation Approach (TRA) to linear systems and introduces, for the first time, an extension of the J-matrix method of scattering to nonlinear phenomena. The TRA provides an algebraic framework for solving second-order linear ordinary differential equations by combining algebraic structure with the analytical properties of orthogonal polynomials and special functions. Computationally, the method benefits from efficient numerical techniques for tridiagonal matrices. Within the TRA, solutions of differential equations, including the Schrödinger equation, are expressed as convergent expansions in square-integrable basis functions chosen to yield a tridiagonal representation of the differential operator. This reduces the problem to a three-term recursion relation for the expansion coefficients, solvable in terms of orthogonal polynomials. For physical systems, key observables such as energy spectra and scattering quantities follow directly from the properties of these polynomials.
The dissertation also examines the J-matrix theory of scattering, where the reference Hamiltonian is treated analytically using a tridiagonal basis representation. Assuming a nonsingular, short-range potential allows truncation within a finite basis subset, with accuracy controlled by basis parameters and size. Unlike variational approaches, the J-matrix method treats the reference Hamiltonian exactly while approximating only the interaction. The J-matrix method has been widely applied in scattering problems in atomic, molecular, and nuclear physics. This work extends its scope to nonlinear phenomena alongside the TRA
analysis of linear systems.
Progressively, we used the TRA to obtain new confined linear systems with a linear energy spectrum. We provided multiple 1D examples and one 3D example to demonstrate the process. We plotted a few wavefunctions of the lowest energy bound states in addition to providing a graphical depiction of the potential function for each confined system. In chapter 4, we treat the inverse problem where for a given choice of the energy spectrum formula E (z2) and a particular collection of basis functions in configuration space { ϕn |(𝑥)}, we provide a general technique for constructing a class of quantum systems associated with the three-parameter continuous dual Hahn orthogonal polynomials Snμ (z2; a, b) . The energy spectrum, the scattering phase shift, and the wavefunction are obtained analytically. However, the corresponding potential function can only be derived numerically and locally for a given set of physical parameters. By choosing the energy as a special function of the polynomial argument E (z2) in chapter 5 and 6 and for the first time in the published literature, we introduce a nonlinear extension of the J-matrix method of scattering. This formulation relies heavily on the linearization formulas for products of orthogonal polynomials. The nonlinear J matrix method (semi analytic) was presented to handle nonlinear Schrodinger equation using a simple toy model in chapter 5 whereas in chapter 6, a perturbative formulation of nonlinear J matrix method in 2D for the NLSE. All results obtained in all cases are new and interesting and they have been published. In the future, as a further research plan, we intend to develop the nonlinear tridiagonal representation approach and the time dependent tridiagonal representation approach.