Conductivity sum rule and super conductivity we write the electrical conductivity as ?(w) = ?' (w) + i?'' (w) where ?', ?" are real.?

(a) Show by a Kramer?s- Kronig relation that

This result is used m the theory of super conductivity. If at very high frequencies (such as x-ray frequencies) ?"(w) is identical for the superconducting and normal states, then we must have but at frequencies 0 nw_g. ?There must be an additional contribution to aj to balance this deficiency.?

(b) Show that if ?'_s (w g) n (w g), as is observed experimentally, then ??_s (w) car1 have a delta function contribution at w = 0, and from the delta function there is a contribution ?''_s (w) = ?'_n?w_g/w. The delta function corresponds to infinite conductivity at zero frequency.?

(c) By elementary consideration of the classical motion of conduction electrons at very high frequencies, show that result found by Ferrell and Glover.

lim wo" (w) 255 o'(s) ds. ["0" (w) dw = f* ; (e) des [*o' (w) dw = mne /2m,