Effective Hamiltonians in a model space

Rather than start from a single nonrelativistic eigenfunction (po and improve it towards the upper component p of the corresponding Dirac spinor ijj, we now start from a d-dimensional model space of nonrelativistic functions (f = 1. d) and we search for an effective Hamiltonian L as a. matrix representation in this model space, such that the eigenvalues of L are equal to a subset of eigenvalues of the Dirac operator. 

To achieve this goal we proceed in some analogy to the two-step approach towards the FW transformation studied in subsection 3.1, however, with a subtle, but important difference. 

The first step consists in a decoupling of electronic from positronic states by means of the not norm-conserving transformation Wa of Eq. (148). At the same step we project to electronic states, since we are only interested in these, i.e. we perform a projecting transformation that transforms from a 4-component spinor space to a 2-component spinor space. 

While the next step for the FW transformation was a renormalization by means of the transformation Wb in order to achieve an overall unitary transformation (which can actually be done before or after the projection to positive energy states), we now apply a transformation that simultaneously block-diagonalizes the Hamiltonian for electrons, such that it isolates a block corresponding to the model space, reestablishes unitarity of this 

Here g is the projector to the model space of nonrelativistic wave functions. What we have done in two steps, can also be done in one step. By means of the operator U we switch from the Dirac operator D — mc to the effective Hamiltonian L in the model space 

---