P-stability, Trigonometric-fitting and the numerical solution of the radial Schrödinger equation

Abstract

In this paper we will study the importance of the properties of P-stability and Trigonometric-fitting for the numerical integration of the one-dimensional Schrödinger equation. This will be done via the error analysis and the application of the studied methods to the numerical solution of the radial Schrödinger equation.

Introduction

The one-dimensional Schrödinger equation can be written as:

$$y^{″}(x)=[l(l+1)/x^2+V(x)-k^2]y(x).$$

The boundary conditions are:

$$y(0)=0$$

and a second boundary condition, for large values of x, determined by physical considerations.

The above boundary value problem occurs frequently in theoretical physics and chemistry, material sciences, quantum mechanics and quantum chemistry, electronics, etc. (see for example [1], [2], [3], [4]).

We can define now some terms in (1):

• The function $W(x)=l(l+1)/x^2+V(x)$ is called the effective potential. This satisfies $W(x)\to 0$ as $x\to \infty$.

• The quantity $E=k^2$ is a real number denoting the energy.

• The quantity l is a given integer representing the angular momentum.

• V is a given function which denotes the potential.

The last years a detailed investigation has been taken place on the development of numerical methods for the solution of the Schrödinger equation. The aim of this research is the development of fast and reliable methods for the solution of the Schrödinger equation and related problems (see for example [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60], [61], [62], [63], [64], [65], [66]).

The methods for the numerical solution of the Schrödinger equation can be divided into two main categories:

• Methods with constant coefficients.

• Methods with coefficients depending on the frequency of the problem.³

The purpose of this paper is to study the importance of the properties of P-stability and Trigonometric-Fitting for the numerical integration of the one-dimensional Schrödinger equation. This will be done via the study of the error analysis and the application of the investigated methods to the numerical solution of the radial Schrödinger equation. We will investigate both categories of methods, i.e.:

• The category of trigonometrically-fitted linear multistep methods and

• The category of P-stable linear multistep methods.

More specifically, we will develop a family of methods of sixth algebraic order for the numerical solution of the radial Schrödinger equation. We will investigate the stability of the two categories of methods. We will study the error of the both categories of methods. Finally, we will apply both categories of methods the new obtained method to the resonance problem. This is one of the most difficult problems arising from the radial Schrödinger equation. The paper is organized as follows. In Section 2 we present the development of the new family of methods. The error analysis is presented in Section 3. In Section 4, we will investigate the stability properties of the new developed methods. In Section 5 the numerical results are presented. Finally, in Section 6 remarks and conclusions are discussed.