Hybrid method (JM-ECS) combining the $J$-matrix and exterior complex scaling methods for scattering calculations

The paper proposes a hybrid method for calculating scattering processes. It combines the $J$-matrix method with exterior complex scaling and an absorbing boundary condition. The wave function is represented as a finite sum of oscillator eigenstates in the inner region, and it is discretized on a grid in the outer region. The method is validated for a one- and a two-dimensional model with partial wave equations and a calculation of $p$-shell nuclear scattering with semirealistic interactions.

Figure 1

The choice of grid points for the finite-difference representation of the Helmholtz equation on an ECS domain. The original problem was stated on $[0,L]$ with outgoing-wave boundary conditions in $L$. The domain is extended with $[L,L^{\text{'}}]$ where a complex grid distance $h\hspace{0.5em}{e}^{i\theta }$ is used. In $L^{\text{'}}$ homogeneous Dirichlet boundary conditions are enforced.

Figure 2

The real and imaginary part of the scattered wave for an Helmholtz problem (1) on the ECS grid. Because of the homogeneous Dirichlet boundary condition the wave is only outgoing in $L$.

Figure 3

The real part of a two-dimensional partial wave of a three-particle $s$-wave scattering problem on an ECS domain (49). The problem has a homogeneous Dirichlet boundary condition on the domain $[0,L^{\text{'}}]{}^2$, and because of the ECS this leads to a problem on the domain $[0,L]{}^2$ with homogeneous Dirichlet boundary condition on the south and west boundaries, and outgoing-wave boundary conditions on the north and east boundaries, where $x=L$ or $y=L$. We show the numerical solution of the problem discretized with finite differences.

Figure 4

This figure illustrates how the kinetic-energy matrix elements are calculated in the last point $R_{n-1}$ of the oscillator representation and in the first point $R_n$ of the finite-difference representation. To calculate $T$ applied on a solution vector we need to translate the oscillator representation to the grid and vice versa.

Figure 5

The real part of the two-dimensional wave function of the three-particle $s$-wave scattering problem of (49) in the hybrid approach with 40 oscillator states and the oscillator parameter $b=0.8$. The values of $X$ and $Y$ in the oscillator region are chosen as the corresponding classical turning points to show the similar behavior of the wave function in different representations. Compare with Fig. (3). Note the jump in the wave function near the $J$-matrix and finite-difference boundary. This jump clearly indicates the different regions in the representation.

Figure 6

There are four regions in the hybrid representation. In region (a) ${\rho }_1,{\rho }_2<R_n$, and the wave function is a sum over product of two oscillator functions. In (b) and (d) one coordinate, respectively, ${\rho }_1<R_n$ and ${\rho }_2<R_n$, is represented in the oscillator state. In region (c) we use finite difference for both coordinates. The amplitude is calculated with the help of a surface integral along the path that is indicated by the solid line a distance $R_k$ away from the axes. Note that between $L$ and Re$(L^{\text{'}})$ the grid is complex valued because of ECS.

Figure 7

$s$-wave phase shift for a Gaussian potential ($V_0=0.2$, $r_0=2$) as a function of wave number $k$ for the problem in (48). Shown are the hybrid JM-ECS results for different matching points $N$ and the VPA (exact) result. The oscillator length is $b=0.3$.

Figure 8

Phase shifts for a Gaussian potential ($V_0=0.2$, $r_0=2$) of Eq. (48) and different angular momenta obtained with JM-ECS (squares), compared with VPA results (solid lines). For all results $b=0.3$ and $N=50$.

Figure 9

The absolute error in phase shift for different sizes of the oscillator basis ($b=0.3$, $l=0$) for the model problem (48).

Figure 10

SCDS results of $s$-wave scattering for model problem (49), for a total energy of 0.471 (solid line $=$ finite differences, dashed line $=$ JM-ECS). The parameters of the calculation are $N=80$ and $b=0.3$. The vertical axis has been rescaled to emphasize the difference.

Figure 11

SCDS results of $s$-wave scattering for model problem (51) with a Gaussian two-body potential (solid line $=$ finite differences, dashed line $=$ JM-ECS). The parameters of the calculation are $N=70$ and $b=1.3$.

Figure 12

Scattering phase shifts for the $\alpha$-$\alpha$ system for different values of total angular momentum calculated with MJM (solid lines) and hybrid JM-ECS method (closed squares). All calculations were made with 80 oscillator states.

Figure 13

Difference between MJM and JM-ECS scattering phases for the 2${}^{+}$ state of the $\alpha$-$\alpha$ system.