Local correlations of different eigenfunctions in a disordered wire

We calculate the correlator of the local density of states $\langle\rho_\varepsilon(\mathbf r_1)\rho_{\varepsilon+\omega}(\mathbf r_2)\rangle$ in quasi-one-dimensional disordered wires in a magnetic field, assuming that $|\mathbf r_1-\mathbf r_2|$ is much smaller than the localization length. This amounts to finding the zero mode of the transfer-matrix Hamiltonian for the supersymmetric $\sigma$ model, which is done exactly by mapping to the three-dimensional Coulomb problem. Both the regimes of level repulsion and level attraction are obtained, depending on $|\mathbf r_1-\mathbf r_2|$. We demonstrate that the correlations of different eigenfunctions in the quasi-one-dimensional and strictly one-dimensional cases are dissimilar.