Multifractality for intermediate quantum systems

TL;DR

This paper proves multifractal scaling laws for arithmetic Seba billiards, connecting quantum multifractality with mathematical methods and zeta function symmetries, advancing the rigorous understanding of quantum intermediate systems.

Contribution

It provides a rigorous mathematical proof of multifractal scaling laws for arithmetic Seba billiards, linking quantum physics concepts with number theory and zeta functions.

Findings

Established multifractal scaling laws for arithmetic Seba billiards.

Derived explicit formulas for fractal exponents in the semiclassical regime.

Connected symmetry relations of fractal exponents to zeta function functional equations.

Abstract

While quantum multifractality has been widely studied in the physics literature and is by now well understood from the point of view of physics, there is little work on this subject in the mathematical literature. I will report on a proof of multifractal scaling laws for arithmetic \u{S}eba billiards. I will explain the mathematical approach to defining the Renyi entropy associated with a sequence of eigenfunctions and sketch how arithmetic methods permit us to obtain a precise asymptotic in the semiclassical regime and how this allows us to compute the fractal exponents explicitly. Moreover, I will discuss how the symmetry relation for the fractal exponent is related to the functional equation of certain zeta functions.