Long-range stiffness of spectral fluctuations in integrable scale-invariant systems

The authors study the semiclassical limit for the Delta 3 statistic of integrable systems which have a homogeneous polynomial as potential. These systems possess a scale invariance which provides one with the energy dependence of the statistic using the general expressions given by Berry (1985). They also obtain the functional dependence of the universal part of the Delta 3 statistic both for integrable and ergodic systems. They use a stationary phase approximation to evaluate the semiclassical limit of the level density when the dimension of the system is larger than one. The investigation of its validity leads to the introduction of a generalised perimeter term in the average level density. The fluctuating part of the semiclassical level density yields the semiclassical limit of the Delta 3 statistic, which is compared numerically to results obtained for actual spectra. They find that the semiclassical approximation is excellent provided the perimeter term is taken into account exactly. They also study the dimensional dependence of Delta 3. For energy levels around the Nth level above the ground state the position of the kink in the Delta 3 statistic is essentially proportional to N(d-1)d/.