Quantum signatures of classical multifractal measures

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A clear signature of classical chaoticity in the quantum regime is the fractal Weyl law, which connects the density of eigenstates to the dimension D₀ 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/absorbed). In the corresponding classical systems D₀ is trivial (equal to the phase-space dimension), and the fractality is manifested in the (multifractal) spectrum of Rényi dimensions Dq. 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 D₀=2 and (ii) the observed dimension oscillates as a function of M and other relevant parameters. We propose a classical model which considers an undersampled measure of the chaotic invariant set, explains our two observations, and predicts that the Weyl law is governed by a non-trivial dimension Dₐsymptotic < D₀ in the semi-classical limit M→∞.