The hole-hole superconducting pairing in t-J model induced by the long-range spin-wave exchange

Long-range $(r \sim p_F^{-1} \gg lattice \ spacing)$ spin-wave exchange produces a very strong pairing of the holes. The different symmetry solutions of BCS-type equation for the superconducting gap $\Delta$ are found. The most strong pairing corresponds to $d$-wave symmetry. The physical reasons for the strong pairing are: 1)large velocity of the spin-wave, 2) a ``collapse'' effect in the attractive potential (-1/r^2) describing the interaction between two holes induced by spin-wave exchange. 3)strongly asymmetric hole dispersion. At concentration of the holes $\delta \sim 0.01-0.3$ we get $\Delta \sim T_c \sim \epsilon_F$. At very low concentration of the holes the many-body wave function of the ground state decays into the product of two-hole bound states.