| Abstract: |
| Are all gaps there? - a question originally posed by Mark Kac for the Almost-Mathieu operator and later termed the Dry Ten Martini Problem by Simon - asks whether all gap labels predicted by the Gap Labelling Theorem are attained by the integrated density of states. For the Almost-Mathieu operator, this problem has been resolved in almost all regimes, and for Sturmian Hamiltonians it was recently answered affirmatively, with all allowed gaps shown to be open.
In this talk, we introduce a geometric analogue of this problem for operators on discrete and metric graphs generated by Sturmian subshifts, where the aperiodicity is encoded in the underlying graph structure rather than in a potential. We study ergodic Jacobi operators on such graphs and compute their gap labels explicitly. In contrast to the classical Sturmian Hamiltonians, we show that not all allowed gaps are open. The missing gaps arise from a geometric mechanism rather than a dynamical one: the local structure of the graph produces flat bands, leading to unattained gap labels. In addition, we observe that spectral bands may touch at special energies, due to a degeneracy in the effective dynamics.
Based on joint work with Ram Band. |
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