The Almost Mathieu Operator (parametrised by two important parameters, $\lambda,\alpha$) has a spectrum that is apparently the Cantor set when $\alpha$ is irratonal and $\lambda \neq 0$. How surprising is this, with respect to `the generic behaviour of operators'? I don't know exactly how to formalise what I mean by generic behaviour, but one route would be something like this:
- For a finite-dimensional Hilbert-space, you can use some random hermitian-matrix ensembles, e.g. with entries drawn from normal distributions, to look at generic behaviour. In the infinite dimensional limit, somehow I am talking about the "limit of these distributions", though I don't know if that is sensible.
- Consider operators on countable-basis Hilbert spaces given by limits of operators defined on finite-dimensional Hilbert spaces (i.e. for the Almost Mathieu Operator, it seems that if you restrict the operator to acting on $|0\rangle,\dots,|n\rangle$ and consider the finite dimensional spectrum and take the limit then the closure, you get the spectrum of the full operator (is this true?))