We consider one dimensional quantum Ising spin-1/2 chains with two-valued nearest neighbor couplings arranged in a quasiperiodic sequence, with uniform, transverse magnetic field. By employing the transfer matrix technique and investigating the dynamics of the corresponding trace map, we show that in the thermodynamic limit the energy spectrum is a Cantor set of zero Lebesgue measure. Moreover, we show that local Hausdorff dimension is continuous and nonconstant over the spectrum. This forms the rigorous counterpart of numerous numerical studies. We also show that the box-counting and the Hausdorff dimensions (both local and global) coincide.
We will be looking at the trace map of the discrete
Schrdinger operator with potential given by the period doubling sequence. It is known that for any positive coupling constant, the spectrum of the corresponding operator is a Cantor set of Lebesgue
measure zero. We are interested in the structure of the spectrum for small coupling constant, specifically the Hausdorff dimension and thickness.