It has proven extremely difficult to describe nuclei directly from a set of nuclear forces, apart from the very lightest elements. In the last decade there has been great progress, supported by computational advances, in various methods to tackle more complex nuclei. Many of these are specialised to systems with even number of neutrons and protons, or just closed shell nuclei, where many calculations simplify. Alternative phenomenological approaches, such as the nuclear shell model, can typically only work within a single major shell.
The complications in the problem arise from the structure of the nuclear force, which is noncentral and spin-dependent, as well as the superfluidity in atomic nuclei. One thus needs to work in large model spaces, take care of the supefluid Cooper pairs, and work with the complex operator structure of the nuclear force. One natural formalism to do all of this is based on the Gor’kov (Green’s function) approach to superfluid systems. Barbieri et al have recently introduced an approach based on the Gor’kov pattern that works for open-shell nuclei , away from the shell closure. A crucial issue for ab initio approaches concerns the ability of performing numerical calculations in increasingly large model spaces, with the aims of thoroughly checking the convergence and of constantly extending the reach to heavier systems.
A long-standing problem with self-consistent calculations of one-body propagators in finite systems concerns the rapid increase of the number of poles generated at each iterative step. The fast growth is expected as the Lehmann representation of one-body Green’s functions develops a continuous cut along the real energy axis in connection with unbound states. This cut is discretised by a growing number of discrete energy states as tthe size of the model space is increased. In practical calculations, one needs to limit the number of discretised poles. We use Lanczos algorithms to project the energy denominators onto smaller Krylov spaces.
As a result of such calculations we can calculate spectral strengths in nuclei that are not easily accessible by other theoretical means. As an example, we show below the spectral strength in 40Ti, with a cut-off ranging up to 13 major shells.