Numerical instabilities

If we try to solve the HFB equations in rather big boxes (typically beyond 30 or 40 fm depending on the mass of the nucleus) we encounter strong numerical instabilities. This problem is not trivially related to the accumulation of errors during the Numerov iteration, because it disappears if the nucleus is treated within the HF approximation instead of HFB. The origin of the problem is in the fact that at large distances the upper and lower HFB wave functions differ by many orders of magnitude, and thus calculations within any number of significant digits must fail at some distance, especially if the small component decreases very fast exponentially. This is so because the small component becomes easily polluted by the large component even if they are coupled by very small matrix elements.

This numerical instability can be removed by neglecting the pairing field beyond a certain large distance $R_{\mathrm{cut}}$. Instead of the full pairing field we consider the truncated field given by

$$\Delta_{\mathrm{cut}}(r)= \left\{ \begin{matrix}\Delta(r) \hfill ... ...m] 0 \hfill & \mbox{for\ } r > R_{\mathrm{cut}}\,. \hfill\\ \end{matrix}\right.$$ (76)

It will be shown in section 8.1 that this truncation has no significant effects on the solution if $R_{\mathrm{cut}}$ is far enough from the origin. Note that by removing the coupling between the lower and upper components one also removes the second term in the asymptotic form of the lower component, see Eq. (36).

Another source of numerical instabilities may come from an accidental degeneracy of deep hole and particle states. In the HFB approach the deep hole states are embedded in the continuum of particle states. The deep hole states are extremely narrow and thus insensitive to the size of the integration box. On the other hand, the discretized particle states are driven by the boundary condition and their spectrum becomes denser as the box is enlarged. As a result, a deep hole state and a quasiparticle (particle like) state can become quasi degenerate and the systematic search for the solution [corresponding to the matching condition (76)] can miss both of them simultaneously. This situation can show up during the iterations and disappear at the final solution; in such a case this numerical accident has no consequence on the final solution. If the quasi degeneracy is met close to the final self-consistent solution, the iterations may not converge. This can be overcome by using a smaller energy step or by slightly changing the size of the integration box in order to lift the degeneracy.