Wave localization in number-theoretic landscapes

We investigate the localization of waves in aperiodic structures that manifest the characteristic multiscale complexity of arithmetic functions with a central role in number theory. In particular, we study the eigenspectra and wave localization properties of tight-binding Schrodinger equation models with onsite potentials distributed according to the Liouville function X(n), the Mobius function mu(n), and the Legendre sequence of quadratic residues modulo a prime. We employ multifractal detrended fluctuation analysis and establish the multifractal scaling properties of the energy spectra in these systems. Moreover, by systematically analyzing the spatial eigenmodes and their level spacing distributions, we show the absence of level repulsion with broadband local-ization across the entire energy spectra. Our study introduces finite-size aperiodic systems whose eigenmodes are all strongly localized and provide opportunities for unique quantum and classical devices of particular importance to cold-atom experiments with engineered speckle potentials as well as unique optical metamaterials and nanostructures with enhanced light-matter interactions.

Automated summary

The study investigates wave localization in aperiodic structures showcasing the multiscale complexity of arithmetic functions. Through analysis of tight-binding Schrodinger equation models, the research reveals the multifractal properties of energy spectra and the widespread localization of spatial eigenmodes.