  Whereas much work in the mathematical literature on quantum chaos has focused
on phenomena such as quantum ergodicity and scarring, relatively little is
known at the rigorous level about the existence of eigenfunctions whose
morphology is more complex. Quantum systems whose dynamics is intermediate
between certain regimes - for example, at the transition between Anderson
localized and delocalized eigenfunctions, or in systems whose classical
dynamics is intermediate between integrability and chaos - have been
conjectured in the physics literature to have eigenfunctions exhibiting
multifractal, self-similar structure. To-date, no rigorous mathematical results
have been obtained about systems of this kind in the context of quantum chaos.
We give here the first rigorous proof of the existence of multifractal
eigenfunctions for a widely studied class of intermediate quantum systems.
Specifically, we derive an analytical formula for the Renyi entropy associated
with the eigenfunctions of arithmetic Seba billiards, in the semiclassical
limit, as the associated eigenvalues tend to infinity. We also prove
multifractality of the ground state for more general, non-arithmetic billiards
and show that the fractal exponent in this regime satisfies a symmetry
relation, similar to the one predicted in the physics literature, by
establishing a connection with the functional equation for Epstein's zeta
function.
