Quantum signatures of classical multifractal measures

A clear signature of classical chaoticity in the quantum regime is the fractal Weyl law, which connects the density of eigenstates to the dimension D-0 of the classical invariant set of open systems. Quantum systems of interest are often partially open (e.g., cavities in which trajectories are partially reflected or absorbed). In the corresponding classical systems D-0 is trivial (equal to the phase-space dimension), and the fractality is manifested in the (multifractal) spectrum of Renyi dimensions D-q. In this paper we investigate the effect of such multifractality on the Weyl law. Our numerical simulations in area-preserving maps show for a wide range of configurations and system sizes M that (i) the Weyl law is governed by a dimension different from D-0 = 2, and (ii) the observed dimension oscillates as a function of M and other relevant parameters. We propose a classical model that considers an undersampled measure of the chaotic invariant set, explains our two observations, and predicts that the Weyl law is governed by a nontrivial dimension D-asymptotic < D-0 in the semiclassical limit M -> infinity.