Two-body Coulomb scattering and complex scaling

The two-body Coulomb scattering problem is solved using the standard complex-scaling method. The explicit enforcement of the scattering boundary condition is avoided. Splitting of the scattering wave function based on the Coulomb modified plane wave is considered. This decomposition leads to a three-dimensional Schrödinger equation with a source term. Partial-wave expansion is carried out and the asymptotic form of the solution is determined. This splitting does not lead to simplification of the scattering boundary condition if complex scaling is invoked. An alternative splitting carried out only on the partial-wave level is introduced and this method is proven to be very useful. The scattered part of the wave function tends to zero at large interparticle distances. This property permits easy numerical solution: the scattered part of the wave function can be expanded on a bound-state-type basis. The method can be applied not only for a pure Coulomb potential but also in the presence of a short-range interaction.

Figure 1

The function ${\omega }_{i,0}(k,r)-{\tau }_0(k,r)$ and the next-to-leading order term of its asymptotic expansion (44) are complex scaled using $\theta =0.1$. Only the real parts are displayed. The momentum is $k=3$, the Sommerfeld parameter is $\gamma =1/3$, and $l=0$. The solid line denotes the exact values and the dashed line corresponds to the asymptotic expansion (44).

Figure 2

The local representation of the phase shift for a pure Coulomb potential (upper panel) and for the general case when a short-range potential is added to the Coulomb term (lower panel). In the first case, the exact solution (dashed black line) is also displayed. For the numerical solution, the boundary condition is imposed at $R=250$ [red (dark gray) line] and $R=1000$ [green (light gray) line]. A detailed discussion is given in the text.

Figure 3

The local representation of the phase shift for a pure Coulomb potential. In the numerical solution, the boundary condition is imposed at $R=250$. The exact solution and the numerical ones are displayed. In the numerical calculations, the number of the Lobatto shape functions $\left(N\right)$ is varied.

Figure 4

The local representation of the phase shift for a pure Coulomb potential as a function of the complex-scaling parameter. Four different $r$ values are considered and, in the local representation of the phase shift, the exact wave function is used.