Semi-spectral Chebyshev method in quantum mechanics

Abstract

Traditionally, finite differences and finite element methods have been by many regarded as the basic tools for obtaining numerical solutions in a variety of quantum mechanical problems emerging in atomic, nuclear and particle physics, astrophysics, quantum chemistry, etc. In recent years, however, an alternative technique based on the semi-spectral methods has focused considerable attention. The purpose of this work is first to provide the necessary tools and subsequently examine the efficiency of this method in quantum mechanical applications. Restricting our interest to timeindependent two-body problems, we obtained the continuous and discrete spectrum solutions of the underlying Schrö dinger or Lippmann-Schwinger equations in both, the coordinate and momentum space. In all of the numerically studied examples we had no difficulty in achieving the machine accuracy and the semi-spectral method showed exponential convergence combined with excellent numerical stability.

Iterative solution of integral equations on a basis of positive energy Sturmian functions

“…Since Sturmians that are eigenfunctions of the general kernel are as difficult to obtain as the solution of the integral equation itself, our method uses auxiliary Sturmian functions, based on the eigenfunctions of a Lippmann-Schwinger integral equation (L-S) using simple auxiliary potentials. The calculations are performed with a spectral expansion into Chebyshev polynomials [20,21], with an accuracy expected to be better than seven to eight significant figures, which is desirable for doing atomic physics calculations. By comparison, the solution of three-body equations for nuclear physics applications, done commonly in momentum space [22], achieve an accuracy not better than four significant figures [23].…”

Modified variable phase method for the solution of coupled radial Schrödinger equations

“…Such an approach was, for example, realized in program [12]. The development of methods for the numerical integration of the Schrödinger equation, aiming at both accuracy and computational efficiency, is still an active subject of investigation (see, for example, [13,14], and references therein). Recently, we developed [15] a novel method for the solution of coupled radial Schrödinger equations, consisting of two steps.…”

Tunneling through a barrier with the phase–amplitude method

“…Seaton and Peach [12] have presented an iterative solution that proved to be very accurate once a spectral expansion [13,14] for the calculation of the amplitude was introduced [15]. However E-mail address: George.Rawitscher@uconn.edu.…”