A. D. Alhaidari

Dirac and Klein–Gordon equations with equal scalar and vector potentials

Abstract We study the three-dimensional Dirac and Klein–Gordon equations with scalar and vector potentials of equal magnitudes as an attempt to give a proper physical interpretation of this class of problems which has recently been accumulating interest. We consider a large class of these problems in which the potentials are noncentral (angular-dependent) such that the equations separate completely in spherical coordinates. The relativistic energy spectra are obtained and shown to differ from those of well-known problems that have the same nonrelativistic limit. Consequently, such problems should not be misinterpreted as the relativistic extension of the given potentials despite the fact that the nonrelativistic limit is the same. The Coulomb, oscillator and Hartmann potentials are considered. Additionally, we investigate the Klein–Gordon equation with uneven mix of potentials leading to the correct relativistic extension. We consider the case of spherically symmetric exponential-type potentials resulting in the S-wave Klein–Gordon–Morse problem.

The rotating Morse potential model for diatomic molecules in the tridiagonal J-matrix representation: I. Bound states

This is the first in a series of articles in which we study the rotating Morse potential model for diatomic molecules in the tridiagonal J-matrix representation. Here, we compute the bound-state energy spectrum by diagonalizing the finite-dimensional Hamiltonian matrix of H2, LiH, HCl and CO molecules for arbitrary angular momentum. The calculation was performed using the J-matrix basis that supports a tridiagonal matrix representation for the reference Hamiltonian. Our results for these diatomic molecules have been compared with available numerical data satisfactorily. The proposed method is handy, very efficient, and it enhances accuracy by combining analytic power with a convergent and stable numerical technique.

Taming the Yukawa potential singularity: improved evaluation of bound states and resonance energies

Using the tools of the J-matrix method, we absorb the 1/r singularity of the Yukawa potential in the reference Hamiltonian, which is handled analytically. The remaining part, which is bound and regular everywhere, is treated by an efficient numerical scheme in a suitable basis using Gauss quadrature approximation. Analysis of resonance energies and bound states spectrum is performed using the complex scaling method, where we show their trajectories in the complex energy plane and demonstrate the remarkable fact that bound states cross over into resonance states by varying the potential parameters.

Relativistic scattering with a spatially dependent effective mass in the Dirac equation

We formulate a relativistic algebraic method of scattering for systems with spatially dependent mass based on the J-matrix method. The reference Hamiltonian is the three-dimensional Dirac Hamiltonian but with a mass that is position-dependent with a constant asymptotic limit. Additionally, this effective mass distribution is locally represented in a finite dimensional function subspace. The spinor couples to spherically symmetric vector and pseudo scalar potentials that are short-range such that they are accurately represented by their matrix elements in the same finite dimensional subspace. We calculate the relativistic phase shift as a function of energy for a given configuration and study the effect of spatial variation of the mass on the energy resonance structure.

Regularization in the J-matrix method of scattering revisited

In three dimensional scattering, the energy continuum wavefunction is obtained by utilizing two independent solutions of the reference wave equation. One of them is typically singular (usually, near the origin of configuration space). Both are asymptotically regular and sinusoidal with a phase difference (shift) that contains information about the scattering potential. Therefore, both solutions are essential for scattering calculations. Various regularization techniques were developed to handle the singular solution leading to different well-established scattering methods. To simplify the calculation the regularized solutions are usually constructed in a space that diagonalizes the reference Hamiltonian. In this work, we start by proposing solutions that are already regular. We write them as infinite series of square integrable basis functions that are compatible with the domain of the reference Hamiltonian. However, we relax the diagonal constraint on the representation by requiring that the basis supports an infinite tridiagonal matrix representation of the wave operator. The hope is that by relaxing this constraint on the solution space a larger freedom is achieved in regularization such that a natural choice emerges as a result. We find that one of the resulting two independent wavefunctions is, in fact, the regular solution of the reference problem. The other is uniquely regularized in the sense that it solves the reference wave equation only outside a dense region covering the singularity in configuration space. However, asymptotically it is identical to the irregular solution. We show that this natural and special regularization is equivalent to that already used in the J-matrix method of scattering.

Representation reduction and solution space contraction in quasi-exactly solvable systems

In quasi-exactly solvable problems partial analytic solutions (energy spectrum and associated wavefunctions) are obtained if some potential parameters are assigned specific values. We introduce a new class in which exact solutions are obtained at a given energy for a special set of values of the potential parameters. To obtain a larger solution space one varies the energy over a discrete set (the spectrum) by simply changing the value of a given integer. A unified treatment that includes the standard as well as the new class of quasi-exactly solvable problems is presented and a few examples are given. The solution space is spanned by discrete square integrable basis functions in which the matrix representation of the Hamiltonian is tridiagonal. Consequently, the wave equation gives a three-term recursion relation for the expansion coefficients of the wavefunction. Imposing quasi-exact solvability constraints results in a complete reduction of the representation to the direct sum of a finite and an infinite component. The finite is real and exactly solvable, whereas the infinite is complex and associated with zero norm states. Consequently, the whole physical space contracts to a finite-dimensional subspace with normalizable states.

The rotating Morse potential model for diatomic molecules in the J-matrix representation: II. The S-matrix approach

This is the second article in which we study the rotating Morse potential model for diatomic molecules using the tridiagonal J-matrix approach. Here, we further improve the accuracy of computing the bound states and resonance energies for this potential model from the poles of the S-matrix for arbitrary angular momentum. The calculation is performed using an infinite square integrable basis that supports a tridiagonal matrix representation for the reference Hamiltonian, which is included in the computations analytically without truncation. Our method has been applied to both the regular and inverted Morse potential with favourable results in comparison with available numerical data. We have also shown that the present method adds a few significant digits to the accuracy obtained from the finite dimensional approach (e.g. the complex rotation method). Moreover, it allows us to easily handle both analytic and non-analytic potentials as well as 1/r singular potentials.

Singular short range potentials in the J-matrix approach

We use the tools of the J-matrix method to evaluate the S-matrix and then deduce the bound and resonance states energies for singular screened Coulomb potentials, both analytic and piecewise differentiable. The J-matrix approach allows us to absorb the 1/r singularity of the potential in the reference Hamiltonian, which is then handled analytically. The calculation is performed using an infinite square integrable basis that supports a tridiagonal matrix representation for the reference Hamiltonian. The remaining part of the potential, which is bound and regular everywhere, is treated by an efficient numerical scheme in a suitable basis using Gauss quadrature approximation. To exhibit the power of our approach we have considered the most delicate region close to the bound-unbound transition and compared our results favorably with available numerical data.

J -matrix method of scattering in any L 2 basis

The restriction imposed on the J-matrix method of using specific

The supersymmetric Jaynes–Cummings model and its solutions

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