Unbounded Trace Orbits of Thue–Morse Hamiltonian

It is well known that, an energy is in the spectrum of Fibonacci Hamiltonian if and only if the corresponding trace orbit is bounded. However, it is not known whether the same result holds for the Thue–Morse Hamiltonian. In this paper, we give a negative answer to this question. More precisely, we construct two subsets Σ _II and Σ _{I II} of the spectrum of the Thue–Morse Hamiltonian, both of which are dense and uncountable, such that each energy in Σ _II∪ Σ _III corresponds to an unbounded trace orbit. Exact estimates on the norm of the transfer matrices are also obtained for these energies: for E∈ Σ _II∪ Σ _III, the norms of the transfer matrices behave like e^c1γ√n≤‖Tn(E)‖≤e^c2γ√n.However, two types of energies are quite different in the sense that each energy in Σ _II is associated with a two-sided pseudo-localized state, while each energy in Σ _III is associated with a one-sided pseudo-localized state. The difference is also reflected by the local dimensions of the spectral measure: the local dimension is 0 for energies in Σ _II and is larger than 1 for energies in Σ _III. As a comparison, we mention another known countable dense subset Σ _I. Each energy in Σ _I corresponds to an eventually constant trace map and the associated eigenvector is an extended state. In summary, the Thue–Morse Hamiltonian exhibits “mixed spectral nature”.