Restricted Friedel sum rules and Korringa relations as consequences of conservation laws

In a general model for impurities in a metallic host it is shown that separate Friedel sum rules hold for the total excess charge, spin and orbital momentum associated with the impurity. These relations are consequences of the conservation laws for charge, spin and orbital magnetic momentum, respectively. Furthermore, the Korringa relations are the exact consequences of the Ward identities, which are, in turn, consequences of some conservation laws in the Anderson model. Thus the Korringa relations are not valid for any of crystalline splitting, spin-orbit interaction and direct electron-electron interaction in the conduction band.