Semiclassical theory of spectral rigidity

Abstract

The spectral rigidity ⊿(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 (ℏ→0), formulae are obtained giving ⊿(L) as a sum over classical periodic orbits. When L ≪Lmax, where Lmax ~ℏ-(N-1) for a system of N freedoms, ⊿(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), ⊿(L)═ 1/5L (as in an uncorrelated (Poisson) eigenvalue sequence); if the system is classically chaotic (all periodic orbits isolated and unstable) and has no symmetry, ⊿(L) ═ In L/2π2+Dif 1≪ L≪ Lmax (as in the gaussian unitary ensemble of random-matrix theory); if the system is chaotic and has time-reversal symmetry, ⊿(L) = In L/π2 + Eif 1 ≪ L ≪ Lmax (as in the gaussian orthogonal ensemble). When L ≫ Lmax, ⊿(L) saturates non-universally at a value, determined by short classical orbits, of order ℏ–(N–1) for integrable systems and In (ℏ-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 ⊿(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).