Consider now, for concreteness, the full Hamiltonian H = H 0 + V with an unperturbed part H 0 made of independent blocks H 0 i . In physics, and in the original motivation for the SW transformation, it is desired that each block corresponds to a distinct energy scale. In particular, all degenerate energy levels should belong to the same block. This well-split Hamiltonian is our starting point H 0 . A perturbative coupling V takes now on a specific meaning: the typical matrix element coupling different sectors must be much smaller than the eigenvalue differences between those sectors. The SW transformation will modify each block H 0 i into an effective Hamiltonian H e f f i incorporating ("integrating out") the effects of the other blocks via the perturbation V . In the end, it is sufficient to look at the sector of interest (called a projection) and to work with the chosen effective Hamiltonian to compute, for instance, eigenvalues and eigenvectors. In physics, this would generate effective low- (or high-)energy Hamiltonians.