Partial Wave Subspace

This is a function in the so-called partial wave subspace (or angular momentum subspace) L ((0,oo),dr) fCmj,Kj- The norm of this function is given by... 

For example, the action of K is just multiplication by the eigenvalue —Kj. The action of the Dirac matrices / and a in the partial wave subspace is described by (110). Likewise, we can compute the action of a spherically symmetric potential in one of the angular momentum subspaces. It remains to observe that due to the factor r in (102) the operator djdr - 1/r in (which is part of expression for the Dirac operator in polar coordinates) simply becomes d/dr in L (0,oo) ... 

In particular, the Dirac operator leaves each of the partial wave subspaces invariant, that is, the result of its action is a wave function in the same angular momentum wave subspace. 

The formal basis employed in the K-matrix calculation includes the relevant partial wave channel (pwc) subspaces plus a "localized channel" (/c) of discrete functions. These last are usual Cl states and their inelusion in the basis allows to efficiently reproduce the autoionizing states and the eorrelation effects. 

In the atomic case, the pwc s are defined by the ion level I and the / value of the electron partial wave, i.e. the formal pwc subsets span the tensor product of the ion level states times the one-electron /-wave manifold. In the following, the subspaces will be indexed with greek letters a subspace index a=/ ,/ will designate explicitly an open pwc subspace, while an index 0 an arbitrary subspace. The Ic subspace will be numbered 0 and Qp will denote the projector in the subspace 0. 

---