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 ≪ L max , where L max ~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 / 5 L (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 /2π 2 + D if 1≪ L ≪ L max (as in the gaussian unitary ensemble of random-matrix theory); if the system is chaotic and has time-reversal symmetry, ⊿( L ) = In L /π 2 + E if 1 ≪ L ≪ L max (as in the gaussian orthogonal ensemble). When L ≫ L max , ⊿( 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).