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).
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