Anderson orthogonality theorem for anisotropic potential

Orthogonality catastrophe is quite well studied, and rather independent from the basis in which it is considered. The point is that the many-particle ground state without the potential (before the potential is turned on) is orthogonal to the ground state with the potential, which, e.g., means that the X-ray absorption cross-section should be zero (the potential is due to the hole left after light absorption), and has wide-reaching implications for the Kondo problem (here the change in the potential is due the flipping of an impurity spin). In most cases the problem is effectively reducible to a one-dimensional one, so anisotropy is not likely to change much - particularly, if approached from the renormalization group viewpoint.

Specific treatments for anisotropic potentials probably exist, but, the information is scattered among the texts on the many_body theory (e.g., Mahan treats it in the context of X-ray absorption), Kondo effect (Bickers review and Anderson's own papers), dephasing/decoherence (e.g., the paper by Aleiner, Wingreen and Meir) bosonization (Schotte&Schotte), etc. I give these not as the definitive references, but as directions to explore.

I think the materials on Kondo effect is where it is most likely to go beyond the s-wave approximation: d- and f-shells of impurities are routinely considered in this context, although this field has also developed many methods alternative to the orthogonality catastrophe approach.