We present a semiclassical approach to $n$-point spectral correlation functions of quantum systems whose classical dynamics is chaotic, for arbitrary $n$. The basic ingredients are sets of periodic orbits that have nearly the same action and therefore provide constructive interference. We calculate explicitly the first correlation functions, to leading orders in their energy arguments, for both unitary and orthogonal symmetry classes. The results agree with corresponding predictions from random matrix theory, thereby giving solid support to the conjecture of universality.

• Received 6 September 2018

DOI:https://doi.org/10.1103/PhysRevE.98.052207