## Abstract

The spectral rigidity $\Delta$(L) of a set of quantal energy levels is the mean square deviation of the spectral staircase from the straight line that best fits it over a range of L mean level spacings. In the semiclassical limit ($\hslash\rightarrow$ 0), formulae are obtained giving $\Delta$(L) as a sum over classical periodic orbits. When $L \simeq L_{max}$, where $L_{max} \sim \hslash^{-(N-1)}$ for a system of N freedoms, $\Delta$(L) is shown to display the following universal behaviour as a result of properties of very long classical orbits: if the system is classically integrable (all periodic orbits filling tori), $\Delta(L) = \frac{1}{15}L$ (as in an uncorrelated (Poisson) eigenvalue sequence); if the system is classically chaotic (all periodic orbits isolated and unstable) and has no symmetry, $\Delta(L) = \ln L/2\pi^2 + D$ if 1 $\simeq L \simeq L_{\max}$ (as in the gaussian unitary ensemble of random-matrix theory); if the system is chaotic and has time-reversal symmetry, $\Delta(L) = \ln L/\pi^2 + E$ if 1 $\simeq L \simeq L_{max}$ (as in the gaussian orthogonal ensemble). When $L \gg L_{max}$, $\Delta(L)$ saturates non-universally at a value, determined by short classical orbits, of order $\hslash^{-(N-1)}$ for integrable systems and $\ln (\hslash^{-1})$ for chaotic systems. These results are obtained by using the periodic-orbit expansion for the spectral density, together with classical sum rules for the intensities of long orbits and a semiclassical sum rule restricting the manner in which their contributions interfere. For two examples $\Delta(L)$ is studied in detail: the rectangular billiard (integrable), and the Riemann zeta function (assuming its zeros to be the eigenvalues of an unknown quantum system whose unknown classical limit is chaotic).

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