Nyström Method for the Coulomb and Screened Coulomb Potentials

Error Estimates for the Nearly Singular Momentum-Space Bound-State Equations

“…where β represents all the other parameters. The usual near singularities (or numerical singularities) are logarithmic near singularity in the screened Coulomb potential and the algebraic near singularity in the screened linear potential and the screened Cornell potential [5][6][7][8]:…”

“…When the near singularities are treated and then the eigenvalue integral Equation 3becomes less nearly singular or free of near singularities [5][6][7][8][16][17][18], the obtained matrix equation corresponding to Equation (32) reads…”

“…If Equation 3is free of near singularities, M ij = 0, ε n = ε n ′ , and ϕ ni = ϕ ni ′ . Let i = n, ϕ nn ′ = 1, and we have from Equation (33) [7,19]…”

“…The odd phenomena (In this paper, the considered odd phenomena arise not from the intrinsic properties of the discussed system but from the inappropriate treatment of the momentumspace bound-state equations.) usually connect with the unreliability of the computed results and serve as an indicator of the unreliability, such as Runge's phenomenon [1,2], the Gibbs phenomenon [3,4], the furcation phenomenon in the numerical eigenfunctions [5][6][7], and the abnormal convergence direction of calculated eigenvalues [8].…”

“…Unlike the common singularities which are explicit and known to us, the near singularities [9][10][11] or numerical sin-gularities [5][6][7][8] are prone to being neglected. Although the discussed problems are not singular analytically in the domain, the near singularity will impair the accuracy of the numerical solutions, sometimes even destroys the reliability of the results.…”