Singular short range potentials in the J-matrix approach

Abstract

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 Jmatrix 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 boundunbound transition and compared our results favorably with available numerical data.

Calculation of information entropies for the 1s2 state of helium‐like ions

“…In terms of three basic parameters, one can define a singular short‐range potential of the form [ 1 ] : where α and β are screening parameters, and A is the strength that is identified with the nuclear charge Z when the potential is used for atomic phenomena. We will assume that F [( α , β ) r ] is bounded everywhere with V ( r ) and has a Coulomb‐like behavior as r → 0 [ie, F (0) = 1].…”

Solutions of Morse potential with position-dependent mass by Laplace transform

“…Then, science from Eqs. (24) and (25) D 0 ¼ Àsb 2 , the energy spectrum can be found by inserting D 0 into (21). In this way, the energy spectrum, yields:…”

“…Several methods have been applied to solve the Schrö-dinger equation, among which are the factorization scheme [15,16], the path integral formulation [17], the supersymmetry approach [18], the algebraic way [19], the power series expansion [20,21], the two-point quasi-rational approximation method [22], the shifted large-N procedure [23], the transfer matrix method [24,25], the asymptotic iteration method [26][27][28], the NikiforovUvarov approach [29][30][31][32], the approximation of perturbation [33] and the auxiliary field method [34].…”

Energy spectrum of a generalized Scarf potential using the asymptotic iteration method and the tridiagonal representation approach

“…The J-matrix method that inspired the TRA, can also give the bound states and resonance energies associated with different short-range potentials. In particular, the approach was used for the Morse potentials [21], the inverse Morse potentials [22], the tamed Yukawa potential [23], the generalized Yukawa potential [24], the Hulthen potential [25], the Hellmann potential [26] and exponential-cosine-screened Coulomb potential [27]. However, there exist a class of non-conventional potentials with discrete spectra and in our work, we seek the solution of one such non-conventional potential in the 1D Schrödinger equation, namely the generalized trigonometric Scarf potential.…”