Sum rule approach to nuclear collective vibration

The NEWSR and the EWSR are studied microscopically within TDA and.RPA methods. By exploiting the analytic properties of the electromagnetic and allowed $\beta$- transitions matrix elements the theory of residue and contour integrals is used to show that exact calculation of the NEWSR and EWSR is possible . In TDA (non -degenerate limit) we deduce formula for the EWSR of $E\lambda$, transition strength, We have given different way of the proof of Thouless theorem about exactly exhausting the EWSR of $E\lambda$ transitions matrix elements in RPA. The unlike particle-hole formalism is developed for allowed p transitions. We find that TDA and RPA conserve the NEWSR and EWSR for Fermi and G-T $\beta$-decays.The NEWSR and the EWSR are studied microscopically within TDA and.RPA methods. By exploiting the analytic properties of the electromagnetic and allowed $\beta$- transitions matrix elements the theory of residue and contour integrals is used to show that exact calculation of the NEWSR and EWSR is possible . In TDA (non -degenerate limit) we deduce formula for the EWSR of $E\lambda$, transition strength, We have given different way of the proof of Thouless theorem about exactly exhausting the EWSR of $E\lambda$ transitions matrix elements in RPA. The unlike particle-hole formalism is developed for allowed p transitions. We find that TDA and RPA conserve the NEWSR and EWSR for Fermi and G-T $\beta$-decays