Correlations in Chaotic Eigenfunctions at Large Separation

An energy eigenfunction in a classically chaotic system is known to have spatial correlations which (in the limit of small $\mathrm{\hslash }$) are governed by a microcanonical distribution in the classical phase space. This result is valid, however, only over coordinate distances which are small compared to any relevant classical distance scales (such as the cyclotron radius for a charged particle in a magnetic field). We derive a modified formula for the correlation function in the regime of large separation. This then permits a complete description, over all length scales, of the statistical properties of chaotic eigenfunctions in the $\mathrm{\hslash }\to 0$ limit. Applications to quantum dots are briefly discussed.