In this survey article it is shown how \(K\)-theory of \(C^*\)-algebras can be used to study the spectral theory of a class of discrete Schrödinger operators \[ H=- \Delta+V \] on \(\ell^ 2(\mathbb{Z})\), where \(-\Delta\) is the discrete Laplacian and \(V\) is a diagonal operator whose diagonal elements are obtained from a substitution sequence. In short, a substitution sequence is defined as follows: Let \(A\) be a finite set of letters, e.g. the set \(\{a,b\}\), and let \(\xi\) be a map that associates to each letter a finite word, e.g. \(\xi(a)=ab\), \(\xi(b)=ba\). A substitution sequence is an infinite sequence of letters, which remains unchanged if each letter in the sequence is replaced by its image under \(\xi\).
Hamiltonians of this type describe quasicrystals. Often they have remarkable spectral properties: The spectrum is singular continuous and supported on a Cantor set of zero Lebesgue measure. Moreover, on the spectral gaps the density of states takes values in some definite countable set of numbers. The density of states \({\mathcal N}: \mathbb{R}_ 0^+\to \mathbb{R}\) is defined by \[ {\mathcal N}(E)= \lim_{\Lambda\nearrow\mathbb{Z}} {1\over {|\lambda|}} N_ \Lambda(E), \] where \(\Lambda= \{-\lambda, -\lambda+1, \dots\lambda\}\) and where \(N_ \Lambda(E)\) denotes the number of eigenvalues smaller or equal to \(E\) of the operator \(H_ \Lambda\), which is the restriction of \(H\) to \(\lambda^ 2(\Lambda)\) with a suitable boundary condition.
The values of \({\mathcal N}\) on the spectral gaps can be used to label these gaps, and the article shows that \(K\)-theory of \(C^*\)-algebras can be used to explain the properties of such Hamiltonians and, in particular, the gap labelling properties of \({\mathcal N}\).