All-coupling frequency independence of Sturmian spectral dimensions

Let d706>0d706>0 and let d6fc(0,1)Qd6fc\in(0,1)\setminus\mathbb{Q} be an irrational frequency. Denote by d6a9λ,αd6a9_{\lambda,\alpha} the phase-independent spectrum of the corresponding Sturmian Schrödinger operator. The paper proves that the Hausdorff and upper box-counting dimensions are almost everywhere constant in d6fcd6fc when d70624d706\geq24. Large-coupling conjecture. The assumption d70624d706\geq24 can be dropped in the theorem, so the same almost-everywhere constancy holds for every d706>0d706>0. A proof is described as currently well out of reach because the methods rely essentially on large coupling.

Sources & referencesView supporting material

Primary source

David Damanik and Anton Gorodetski, “Almost Sure Frequency Independence of the Dimension of the Spectrum of Sturmian Hamiltonians”, arXiv:1406.4810 (2014).

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Solutions 0

No solutions have been posted yet.