### The Dry Ten Martini Problem for Sturmian Schrödinger Operators

Mathematical Physics Seminar

20th September 2024, 2:00 pm – 3:00 pm

Fry Building, 2.04

"Are all gaps there?", asked Mark Kac in 1981 during a talk at the AMS annual meeting, and offered ten Martinis for the answer. This led Barry Simon to coin the names the Ten Martini Problem (TMP) and the Dry Ten Martini Problem for two related problems concerning the Almost-Mathieu operator. The TMP is about showing that the spectrum of the Almost-Mathieu operator is a Cantor set. The Dry TMP is about the values that the integrated density of states (IDS) attains at the spectral gaps. The gap labelling theorem predicts the possible set of values which the IDS may attain at the spectral gaps. The Dry TMP is whether or not all these values are attained, or equivalently, "are all gaps there?".

We present an affirmative solution to the Dry Ten Martini Problem for Sturmian Hamiltonians. Concretely, it is proved that all spectral gaps are there for Schrödinger operators with Sturmian potentials and non-vanishing coupling constant. The talk is based on a joint work with Siegfried Beckus and Raphael Loewy.

*Organiser*: Thomas Bothner

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