"Mixed spectral nature" of Thue--Morse Hamiltonian

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".