Multifractality of Semiclassical Measures on Star Graphs

Abstract

We study eigenfunctions of quantum star graphs in the large edge number limit through the edge-mass distributions associated with their semiclassical measures. For generic edge lengths, we show that these distributions can realize every admissible multifractal scaling law along suitable subsequences of eigenvalues. We also prove a constructive result for quasi-equilateral star graphs. Starting from prescribed probability measures, we construct graphs and locate eigenvalues inside spectral clusters whose eigenfunctions reproduce the same scaling behavior. These results show that quantum star graphs form an explicit model realizing the full range of admissible multifractal behavior between localization and equidistribution.