Are all possible spectral gaps, predicted by the Gap labelling theorem, open for a given Schrödinger operator? This is the so called ”Dry Ten Martini problem (Dry TMP)” motivated 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 frequencies 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.