Friedel-type sum rule for a general periodic potential

By using the continual direct-sum decomposition obtained previously for the t operators, it is shown that the change of the density of states in the presence of a general (non-muffin-tin) periodic potential can be given in a determinant form involving only the on-shell t matrix. The result obtained is close to the usual Friedel sum rule of scattering theory which, however, relies strongly on the existence of asymptotic free motion and hence, on the existence of the phase-shift motion. Thus, since the periodic potential does not allow for the existence of asymptotic free motion, the present result illustrates a rather general trace property of the resolvent, which appears to be independent of the very different boundary conditions in the two cases.