AG Mathematische Physik, Ram Band: The Dry Ten Martini Problem for Sturmian Schrödinger Operators

Ram Band

The Dry Ten Martini Problem for Sturmian Schrödinger Operators

Abstract:

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 – this is proven in a joint work with Siegfried

Beckus and Raphael Loewy.

In a joint work with Gilad Sofer, we study families of Sturmian

operators on metric and discrete graphs.

We show how their gap labels are characterized and what would be the

corresponding Dry Ten Martini Problem for those.