Geometry of effective Hamiltonians

We give a complete geometrical description of the effective Hamiltonians common in nuclear shell-model calculations. By recasting the theory in a manifestly geometric form, we reinterpret and clarify several points. Some of these results are hitherto unknown or unpublished. In particular, commuting observables and symmetries are discussed in detail. Simple and explicit proofs are given, and numerical algorithms are proposed that improve and stabilize methods commonly used today.

Figure 1

Plane spanned by $|{\xi }_k\rangle$ and $|{\eta }_k\rangle$, and action of projectors $P$ and $Q$ on $|{\eta }_k\rangle$.