Gutzwiller's trace formula and spectral statistics: Beyond the diagonal approximation

We calculate the 2-point spectral correlation function for classically chaotic systems in the semiclassical limit using Gutzwiller's trace formula. The off-diagonal contributions from pairs of nonidentical periodic orbits are evaluated by relating them to the diagonal terms. The behavior we find is similar to that recently discovered to hold for disordered systems using nonperturbative supersymmetric methods. Our analysis generalizes immediately to include parametric statistics and higher-order correlations and to the study of the semiclassical distribution of matrix elements.