2021/03/31 by Jonathan P. Keating, Henrik Ueberschaer
Mathematics · Physics and Astronomy · #Dynamical billiards #Eigenfunction #Eigenvalues and eigenvectors #Fractal #Harmonic oscillator #Mathematical analysis #Mathematical physics #Mathematics #Multifractal system #Physics #Quantum #Quantum chaos #Quantum chaos and dynamical systems #Quantum dynamics #Quantum many-body systems #Quantum mechanics #Quasiperiodic function #Semiclassical physics #Statistical physics #Theoretical and Computational Physics #math-ph #math.MP #msc:35P05 #msc:81Q50
paper · pdf · doi:10.1007/s00220-021-04214-8
published in Communications in Mathematical Physics 389(1), 543-569 (Springer Science+Business Media) · 28 pages, to appear in Comm. Math. Phys
openalex publication_date 2021/11/18 · arxiv created 2022/02/28 · arxiv updated 2022/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
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.