Periodic orbit resummation and the quantization of chaos

We study the spectral determinant ∆(E), which has, by construction, zeros at the quantum energy levels of a given system. If the classical motion of the system in question is chaotic then ∆(E) has a semiclassical representation as a sum over combinations of periodic orbits. There are, however, a number of fundamental problems associated with its convergence properties. Imposing upon the sum the condition that, like ∆(E) itself, it is real for real E, we obtain formal resummation equations relating the contributions from asymptotically long orbits to those of the short orbits. These then lead to a formal derivation of the previously conjectured ‘Riemann-Siegel lookalike’ formula, which involves only a finite orbit sum and thus represents, in principle, a semiclassical rule for quantizing chaos.