"Mixed spectral nature" of Thue--Morse Hamiltonian

TL;DR

This paper investigates the spectral properties of the Thue-Morse Hamiltonian, revealing a complex mix of extended, pseudo-localized, and one-sided pseudo-localized energies, with detailed estimates on transfer matrix norms and spectral measure dimensions.

Contribution

It identifies three dense spectral subsets with distinct behaviors and provides precise estimates on transfer matrix norms and local spectral dimensions, highlighting the mixed spectral nature.

Findings

Three dense spectral subsets with different properties

Transfer matrix norms grow like e^{c√n} for certain energies

Spectral measure local dimension varies across subsets

Abstract

We find three dense subsets $\Sigma_I,\Sigma_{II}$ and $\Sigma_{III}$ of the spectrum of the Thue-Morse Hamiltonian, such that each energy in $\Sigma_I$ is extended, each energy in $\Sigma_{II}$ is pseudo-localized and each energy in $\Sigma_{III}$ is one-sided pseudo-localized. We also obtain exact estimations on the norm of the transfer matrices and the norm of the formal solutions for these energies. Especially, for $E\in \Sigma_{II}\cup\Sigma_{III}, $ the norms of the transfer matrices behave like $$ e^{c_1\gamma\sqrt{n}}\le\|T_{n}(E)\|\le e^{c_2\gamma\sqrt{n}}. $$ The local dimensions of the spectral measure on these subsets are also studied. The local dimension is $0$ for energy in $\Sigma_{II}$ and is larger than $1$ for energy in $\Sigma_{I}\cup\Sigma_{III}.$ In summary, the Thue-Morse Hamiltonian exhibits "mixed spectral nature".