Continuity properties of the electronic spectrum of 1D quasicrystals (Q1181849)

\textit{Ostlund} and \textit{Kim} numerically computed the spectrum of the hamiltonian defined on \(\mathbb{C}^ 2(Z)\) by \[ H(\alpha,x)\psi(n)=\psi(n+1)+\psi(n-1)+\chi_{[1- \alpha,1[}(x+n\alpha), \qquad \alpha,x\in[0,1[, \] for rational values of \(\alpha\) [Physica Scripta 9, 193-198 (1985)]. The plot of energy as a function of \(\alpha\) has a fractal nature. This paper gives a mathematical explanation of the Ostlund-Kim spectrum. The authors have proved for more general hamiltonians that the map \(\alpha\in[0,1[\to G_ \alpha\) is continuous at irrational values of \(\alpha\) and discontinuous at rational numbers. The results are obtained through \(C^*\)-algebra techniques. Such models can describe the phonon spectra in a one dimensional quasicrystal.