Dry Ten Martini Problem for Sturmian Dynamical Systems

Thursday 10th October 16:00-17:00 Maths 311B

Are all possible spectral gaps, predicted by the Gap labelling theorem, open for a given

Schr¨odinger operator? This is the so called ”Dry Ten Martini problem (Dry TMP)” moti

vated by the ”Ten Martini Problem (TMP)”. The name TMP was coined by Barry Simon

after Mark Kac offered in 1981 ten Martinis to anyone who solves it. Originally, the TMP

was proposed for the Almost Mathieu operator conjecturing Cantor spectrum for all couplings

and all irrational frequencies. The TMP for the Almost Mathieu operator was solved by Artur

Avila and Svetlana Jitomirskaya in 2005.

In this talk, we discuss the Dry TMP for so-called Sturmian dynamical systems. These systems

define a one-dimensional Schroedinger operator where the potential is characterized in terms

of two parameters: a frequency paramter and strength of the coupling constant. Like for the

Almost-Mathieu operator one asks if all predicted spectral gaps are open for all irrational fre

quencies and all couplings. For large couplings, the Dry TMP for Sturmian systems was solved

by Raymond in 1997. In 2016, the Inventiones paper by David Damanik, Anton Gorodetski

and William Yessen provided a solution if the frequency is the golden mean for all non-zero

couplings.

In a current project with Ram Band and Raphael Loewy we solve the Dry TMP for all irrational

frequencies and all couplings by a detailed control of suitable periodic approximations. In the

talk we present the problem and the route to its resolution.

We use cookies