The partial-wave subspaces

Consider two functions / (r) and / (r) that are square-integrable on the interval 0 r oo. We can use them to define a function 

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

Making the transition back to cartesian coordinates, we obtain a square inte-grable function (x) = (l/r)t/ (r, d, (f) (whose norm is again given by the above expression). An arbitrary function is a linear combination of the form 

We put everything together and obtain the following result Consider the Dirac operator with the potential 

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. 

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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) ... 

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