Regularity of the spectrum for the almost Mathieu operator (Q2706581)

Denote by \(\mu\) the integrated density of states of the almost Mathieu operator with (irrational) frequency \(\alpha\) and coupling constant \(\beta\). It is proved that NEWLINE\[NEWLINE\int\log|z-s|d\mu(s)= \max\{0, \log|\beta|\}NEWLINE\]NEWLINE if \(z\) belongs to the spectrum \(\text{Sp}(\alpha, \beta)\) and \(\alpha\) satisfies a certain Diophantine condition. This equality implies in particular that \(\text{Sp}(\alpha, \beta)\) is a regular compactum and \(\mu\) is its equilibrium distribution. Moreover, the concept of a (in-)complete scale of function algebras is introduced and a generic construction is discussed that may be used to extend the above-mentioned result to a more general class of parameters \(\alpha\), \(\beta\).