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Multifractal eigenfunctions for quantum star graphs

We prove that the eigenfunctions of quantum star graphs exhibit multifractal self-similar structure in certain specified circumstances. In the semiclassical regime, when the spectral parameter and the number of vertices tend to infinity, we derive an asymptotic condition for the Mellin transform of a specified function arising from the set of bond lengths which yields an asymptotic for the Renyi entropy associated with an eigenfunction. We apply this result to show that one may construct simple quantum star graphs which satisfy a multifractal scaling law. In the low frequency regime we prove multifractality by computing the Renyi entropy in terms of a zeta function associated with the set of bond lengths. In certain arithmetic cases the fractal exponent D_q satisfies a symmetry relation around q=1/4 which arises from the functional equation of the zeta function. Our results are, in some sense, analogous to the multifractal scaling law that the authors recently proved for arithmetic Seba billiards. However, unlike in that case, we do not require arithmetic conditions to be satisfied, and nor do we rely on delicate arithmetic estimates.