Multifractal eigenfunctions for a singular quantum billiard

TL;DR

This paper rigorously proves the existence of multifractal eigenfunctions in intermediate quantum systems, specifically for arithmetic Seba billiards, revealing complex self-similar structures in their eigenstates.

Contribution

It provides the first rigorous mathematical proof of multifractality in eigenfunctions for a class of intermediate quantum billiards, connecting fractal exponents with Epstein's zeta function.

Findings

Derived an analytical formula for the Renyi entropy of eigenfunctions

Proved multifractality of the ground state in non-arithmetic billiards

Established a symmetry relation for the fractal exponent

Abstract

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.