Brief Survey on the CP Methods for the Schrödinger Equation

Abstract

The CP methods have some salient advantages over other methods, viz.: (i) the accuracy is uniform with respect to the energy E; (ii) there is an easy control of the error; (iii) the step widths are unusually big and the computation is fast; (iv) the form of the algorithm allows a direct evaluation of collateral quantities such as normalisation constant, Prüfer phase, or the derivative of the solution with respect to E; (v) the algorithm is of a form which allows using parallel computation.

Appendix

Functions ξ(Z),η ₀(Z),η ₁(Z),…, originally introduced in [9] (they are denoted there as $\bar{\xi}(Z),\bar{\eta}_{0}(Z),\bar{\eta}_{1}(Z),\ldots)$, are defined as follows:

$$\xi (Z)=\left\{ \begin{matrix} \cos (|Z{{|}^{{1}/{2}\;}})\,\,if\,Z\le 0, \\ \cosh ({{Z}^{{1}/{2}\;}})\,\,if\,Z>0, \\ \end{matrix} \right.\ $$

(A.1)

$${{\eta }_{0}}(Z)=\left\{ \begin{matrix} {\sin (|Z{{|}^{{1}/{2}\;}})}/{|Z{{|}^{{1}/{2}\;}}}\;\,\,\,\,\,if\,Z<0, \\ \begin{matrix} 1 & if\,Z=0, \\ \end{matrix} \\ \begin{matrix} {\sinh ({{Z}^{{1}/{2}\;}})}/{{{Z}^{{1}/{2}\;}}}\; & if\,Z>0, \\ \end{matrix} \\ \end{matrix} \right.\ $$

(A.2)

η ₁(Z),η ₂(Z),…, are constructed by recurrence:

$$\begin{aligned}[c]\eta_1(Z)&=[\xi(Z)-\eta_0(Z)]/Z,\\\eta _m&=[\eta_{m-2}(Z)-(2m-1)\eta_{m-1}(Z)]/Z,\quad m=2,3,\ldots.\end{aligned}$$

(A.3)

Some useful properties are as follows:

• (i) Series expansion:

  $$\eta_m(Z)=2^m\sum_{q=0}^\infty{g_{mq}Z^q\over (2q+2m+1)!},$$

  (A.4)

  with

  $$g_{mq}=\begin{cases}1&\mbox{if}\ m=0,\\(q+1)(q+2)\cdots(q+m)&\mbox{if}\ m>0 .\end{cases}$$

  (A.5)

  In particular,

  $$\eta_m(0)={1\over(2m+1)!!},$$

  (A.6)

  where (2m+1)!!=1×3×5×⋅⋅⋅×(2m+1).

• (ii) Asymptotic behaviour at large |Z|:

  $$\eta_m(Z)\approx \begin{cases}\xi(Z)/Z^{(m+1)/2}&\mbox{for odd}\ m,\\\eta_0(Z)/Z^{m/2}&\mbox{for\ even}\ m .\end{cases}$$

  (A.7)

• (iii) Differentiation properties:

  $$\xi'(Z)={1\over2}\eta_0(Z),\qquad\eta'_m(Z)={1\over2}\eta_{m+1}(Z),\quad m=0,1,2,\ldots.$$

  (A.8)