Another Martini for quantum graphs
Seminar Room 1, Newton Institute
I will report on joint work with V. Geyler and K. Pankrashkin. As one might guess from the title, the topic of the talk refers to the "Ten Martini Problem" which consisted in determining the set theoretic nature - band or Cantor structure - of the Harper operator in terms of number theoretic properties of a certain parameter. We treat the analogous problem for a periodic square quantum graph with magnetic field and suitable boundary conditions. We achieve the desired characterization of the spectrum essentially by reduction to a discrete periodic case for which the answer was given recently by Avila and Jitomirskaya.