Staircase functions, spectral rigidity, and a rule for quantizing chaos

Considering the Selberg trace formula as an exact version of Gutzwiller’s semiclassical periodic-orbit theory in the case of the free motion on compact Riemann surfaces with constant negative curvature (Hadamard-Gutzwiller model), we study two complementary basic problems in quantum chaology: the computation of the classical staircase N(l), the number of periodic orbits with length shorter than l, in terms of the quantal energy spectrum {${\mathit{E}}_{\mathit{n}}$}; the computation of the spectral staircase scrN(E), the number of quantal energies below the energy E, in terms of the length spectrum {${\mathit{l}}_{\mathit{n}}$} of the classical periodic orbits. A formulation of the periodic-orbit theory is presented that is intrinsically unsmoothed, but for which an effective smoothing arises from the limited ‘‘input data,’’ i.e., from the limited knowledge of the periodic orbits in the case of scrN(E) and the limited knowledge of quantal energies in the case of N(l). Based on the periodic-orbit formula for scrN(E), we propose a rule for quantizing chaos, which simply states that the quantal energies are determined by the zeros of the function ${\xi }_1$(E)=cos[πscrN(E)]. The formulas for N(l) and scrN(E) as well as the quantization condition are tested numerically. Furthermore, it is shown that the staircase scrN(E) computed from the length spectrum yields (up to a constant) a good description of the spectral rigidity ${\mathrm{\Delta }}_3$(L), and thus we are able to compute a statistical property of the quantal energy spectrum of a chaotic system from classical periodic orbits.