Tridiagonal Representation Approach Research Articles

Revisiting the Coulomb problem: A novel representation of the confluent hypergeometric function as an infinite sum of discrete Bessel functions

We use the tridiagonal representation approach to solve the radial Schrödinger equation for the continuum scattering states of the Coulomb problem in a complete basis set of discrete Bessel functions. Consequently, we obtain a new representation of the confluent hypergeometric function as an infinite sum of Bessel functions, which is numerically very stable and more rapidly convergent than another well-known formula.

Electrostatic multipole contributions to the binding energy of electrons

The interaction of an electron with a local static charge distribution (e.g., an atom or molecule) is dominated by the radial 1/r Coulomb potential at large distances. The second-order effect is generated by the non-central electric dipole contribution cosθ/r2. The third-order effect is attributed to the electric quadrupole potential (3cos2θ-1)/2r3. In this study, we used the tridiagonal representation approach to produce a reasonably accurate account for the combined effects of every contribution to the binding energy of the electron with an effective quadrupole interaction. As a result, we obtained the bound states of a valence electron in an atom with both electric dipole and quadrupole moments.

Progressive approximation of bound states by finite series of square-integrable functions

We use the “tridiagonal representation approach” to solve the time-independent Schrödinger equation for bound states in a basis set of finite size. We obtain two classes of solutions written as a finite series of square integrable functions that support a tridiagonal matrix representation of the wave operator. The differential wave equation becomes an algebraic three-term recursion relation for the expansion coefficients of the series, which is solved in terms of finite polynomials in the energy and/or potential parameters. These orthogonal polynomials contain all physical information about the system. The basis elements in configuration space are written in terms of either the Romanovski–Bessel polynomial or the Romanovski–Jacobi polynomial. The maximum degree of both polynomials is limited by the polynomial parameter(s). This makes the size of the basis set finite but sufficient to give a very good approximation of the bound state wavefunctions that improves with an increase in the basis size.

Finite series representation for the bound states of a spiked isotropic oscillator with inverse-quartic singularity

In this paper, we use the tridiagonal representation approach to obtain a finite series solution of the radial Schrödinger equation in three dimensions for a spiked oscillator with inverse quartic singularity.

Bound states for generalized trigonometric and hyperbolic Pöschl–Teller potentials

We use the “tridiagonal representation approach” to solve the time-independent Schrödinger equation for the bound states of generalized versions of the trigono-metric and hyperbolic Pöschl–Teller potentials. These new solvable potentials do not belong to the conventional class of exactly solvable problems. The solutions are finite series of square integrable functions written in terms of the Jacobi polynomial.

Deformed Morse-like potential

We introduce an exactly solvable one-dimensional potential that supports both bound and/or resonance states. This potential is a generalization of the well-known 1D Morse potential where we introduced a deformation that preserves the finite spectrum property. On the other hand, in the limit of zero deformation, the potential reduces to the exponentially confining potential well introduced recently by Alhaidari [Theor. Math. Phys. 206, 84–96 (2021)]. The latter potential supports an infinite spectrum, which means that the zero deformation limit is a critical point where our system will transition from the finite spectrum limit to the infinite spectrum limit. We solve the corresponding Schrodinger equation and obtain the energy spectrum and the eigenstates using the tridiagonal representation approach.

Erratum to: Exponentially confining potential well

We introduce an exponentially confining potential well that could be used as a model to describe the structure of a strongly localized system. We obtain an approximate partial solution of the Schr\"odinger equation with this potential well where we find the lowest energy spectrum and corresponding wavefunctions. The Tridiagonal Representation Approach (TRA) is used as the method of solution, which is obtained as a finite series of square integrable functions written in terms of the Bessel polynomial.

Solutions of a Bessel-type differential equation using the Tridiagonal Representation Approach

We obtain a class of exact solutions of a Bessel-type differential equation, which is a six-parameter linear ordinary differential equation of the second order with irregular (essential) singularity at the origin. The solutions are obtained using the Tridiagonal Representation Approach (TRA) as bounded series of square integrable functions written in terms of the Bessel polynomial on the real line. The expansion coefficients of the series are orthogonal polynomials in the equation parameters space. We use our findings to obtain solutions of the Schr\"odinger equation for some novel potential functions.

Comment on “Bound states and the potential parameter spectrum” [J. Math. Phys. 61, 062103 (2020)

We analyze the application of the “tridiagonal representation approach” (TRA) to the Schrödinger equation for some simple, exactly solvable, quantum-mechanical models. In the case of the Kratzer–Fues potential, the expression for the energy does not appear to be correct. In addition, the well known Frobenius method, which resembles the TRA, is far simpler, clearer, and more elegant, in addition to giving the correct result.

Bound states of a quartic and sextic inverse-power-law potential for all angular momenta

We use the tridiagonal representation approach to solve the radial Schrödinger equation for an inverse-power-law potential of a combined quartic and sextic degrees and for all angular momenta. The amplitude of the quartic singularity is larger than that of the sextic but the signs are negative and positive, respectively. It turns out that the system has a finite number of bound states, which is determined by the larger ratio of the two singularity amplitudes. The solution is written as a finite series of square integrable functions written in terms of the Bessel polynomial.

Exponentially confining potential well

We introduce an exponentially confining potential well that can be used as a model to describe the structure of a strongly localized system. We obtain an approximate partial solution of the Schrodinger equation with this potential well where we find the lowest energy spectrum and the corresponding wavefunctions. We use the tridiagonal representation approach as the method for obtaining the solution as a finite series of square-integrable functions written in terms of Bessel polynomials.

Treatment of a three-dimensional central potential with cubic singularity

We compute the bound states for a special type of singular central potential that generalizes the hyperbolic Eckart potential by adding a cubic singular term at the origin while keeping the short range exponential decay far away from the origin. Such strong singular potentials are of practical importance in atomic, nuclear and molecular physics. To bring the solution of the Schrodinger equation for finite angular momentum to analytical treatment we use an analytical approximation to the centrifugal orbital part of the potential that has a similar structure to the Eckart potential. We compute the energy spectrum associated with this potential using both the tridiagonal representation approach (TRA) and the asymptotic iteration method (AIM) and make a comparative analysis of these results.

The energy spectrum of a new exponentially confining potential

In this work, we consider a general Morse-type confining potential that was introduced recently by Alhaidari (J Theor Math Phys 205, 2020. arXiv:2005.09080 ). This potential is completely confining and hence has an infinite discrete spectrum. We compute the energy spectrum associated with this confining potential using two different approaches so as to ensure the correctness of our results. We use both asymptotic iteration method and the numerical diagonalization method based on the tridiagonal representation approach of our unperturbed Hamiltonian to compute the energy spectrum. We found that both approaches resulted in bound state energies that are in agreement with each other to a high degree of accuracy.

Bound states and the potential parameter spectrum

In this article, we answer the following question: If the wave equation possesses bound states, but it is exactly solvable for only a single non-zero energy, can we find all bound state solutions (energy spectrum and associated wavefunctions)? To answer this question, we use the “tridiagonal representation approach” to solve the wave equation at the given energy by expanding the wavefunction in a series of energy-dependent square integrable basis functions in configuration space. The expansion coefficients satisfy a three-term recursion relation, which is solved in terms of orthogonal polynomials. Depending on the selected energy, we show that one of the potential parameters must assume a value from within a discrete set called the “potential parameter spectrum” (PPS). This discrete set is obtained from the spectrum of the above polynomials and can be either a finite or an infinite set. Inverting the relation between the energy and the PPS gives the bound state energy spectrum. Therefore, the answer to the above question is affirmative.

Series Solution of a Ten-Parameter Second-Order Differential Equation with Three Regular Singularities and One Irregular Singularity

We introduce a ten-parameter ordinary linear differential equation of the second order with four singular points. Three of these are finite and regular whereas the fourth is irregular at infinity. We use the tridiagonal representation approach to obtain a solution of the equation as bounded infinite series of square integrable functions that are written in terms of the Jacobi polynomials. The expansion coefficients of the series satisfy three-term recursion relation, which is solved in terms of a modified version of the continuous Hahn orthogonal polynomial.

Bound states of an inverse-cube singular potential: A candidate for electron-quadrupole binding

We use the Tridiagonal Representation Approach (TRA) to obtain exact bound states solution (energy spectrum and wave function) of the Schrödinger equation for a three-parameter short-range potential with [Formula: see text], [Formula: see text] and [Formula: see text] singularities at the origin. The solution is a finite series of square-integrable functions with expansion coefficients that satisfy a three-term recursion relation. The solution of the recursion is a non-conventional orthogonal polynomial with discrete spectrum. The results of this work could be used to study the binding of an electron to a molecule with an effective electric quadrupole moment which has the same [Formula: see text] singularity.

Tridiagonal representation approach in quantum mechanics

We present an algebraic approach for finding exact solutions of the wave equation. The approach, which is referred to as the tridiagonal representation approach, is inspired by the J-matrix method and based on the theory of orthogonal polynomials. The class of exactly solvable problems in this approach is larger than the conventional class. All properties of the physical system (energy spectrum of the bound states, phase shift of the scattering states, energy density of states, etc) are obtained in this approach directly and simply from the properties of the associated orthogonal polynomials.

Electric dipole and quadrupole contributions to valence electron binding in a charge-screening environment

We make a multipole expansion of the atomic/molecular electrostatic charge distribution as seen by the valence electron up to the quadrupole term. The Tridiagonal Representation Approach (TRA) is used to obtain an exact bound state solution associated with an effective quadrupole moment and assuming that the electron-molecule interaction is screened by suborbital electrons. We show that the number of states available for binding the electron is finite forcing an energy jump in its transition to the continuum that could be detected experimentally in some favorable settings. We expect that our solution gives an alternative, viable and simple description of the binding of valence electron(s) in atoms/molecules with electric dipole and quadrupole moments. We also ascertain that our model implies that a pure quadrupole-bound anion cannot exist in such a charge-screening environment.

Exact scattering and bound states solutions for novel hyperbolic potentials with inverse square singularity

We use the Tridiagonal Representation Approach (TRA) to obtain exact scattering and bound states solutions of the Schrödinger equation for short-range inverse-square singular hyperbolic potentials. The solutions are series of square integrable functions written in terms of the Jacobi polynomial with the Wilson polynomial as expansion coefficients. The series is finite for the discrete bound states and infinite but bounded for the continuum scattering states.

Exact solvability of two new 3D and 1D non-relativistic potentials within the TRA framework

This work aims at introducing two new solvable 1D and 3D confined potentials and present their solutions using the Tridiagonal Representation Approach (TRA). The wave function is written as a series in terms of square integrable basis functions which are expressed in terms of Jacobi polynomials. The expansion coefficients are then written in terms of new orthogonal polynomials that were introduced recently by Alhaidari, the analytical properties of which are yet to be derived. Moreover, we have computed the numerical eigenenergies for both potentials by considering specific choices of the potential parameters.