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Theory on Superconducting Transition from Pseudogap State

The anomalous properties of High-$T_{\rm c}$ cuprates are investigated both in the normal state and in the superconducting state. In particular, we pay atte ntion to the pseudogap in the normal state and the phase transition from the pse udogap state to the superconducting state. The pseudogap phenomena observed in cuprates are naturally understood as a precursor of the strong coupling superconductivity. We have previously show n by using the self-consistent T-matrix calculation that the pseudogap is a resu lt of the strong superconducting fluctuations which are accompanied by the stron g coupling superconductivity in quasi-two dimensional systems [J. Phys. Soc. Jpn . {\bf68} (1999) 2999.]. We extend the scenario to the superconducting state. The close relation between the pseudogap state and the superconducting state is pointed out. Once t he superconducting phase transition occurs, the superconducting order parameter rapidly grows rather than the result of BCS theory. With the rapid growth of the order parameter, the gap structure becomes sharp, while it is remarkably broad in the pseudogap state. The characteristic energy scale of the gap does not chan ge. These results well explain the phase transition observed in the spectroscopi c measurements. Further, we calculate the magnetic and transport properties which show the pseu dogap phenomena. The comprehensive understanding of the NMR, the neutron scatter ing, the optical conductivity and the London penetration depth is obtained both in the pseudogap state and in the superconducting state.