Correlations of Eigenfunctions in Disordered Systems

Correlations of eigenfunctions, $\langle|\psi_k(r_1)|^2|\psi_l(r_2)|^2\rangle$, in a disordered system are investigated. We derive general formulae expressing these correlation functions in terms of the supermatrix sigma-model. In particular case of weak localization regime we find that the correlations of the same eigenfunction are proportional to $g^{-1}$ for large distances, while the correlations of two different eigenfunctions cross over from $g^{-1}$ behavior for $r_1=r_2$ to $g^{-2}$ one for $\vert r_1 - r_2 \vert \gg l$, with $g$ and $l$ being the dimensionless conductance and the mean free path, respectively.