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One-dimensional quasi-periodic Schrödinger operators III

Thursday 15th January 2015 - 11:30 to 12:30
INI Seminar Room 1
Quasi-periodic Schrödinger operators arise in solid state physics, describing the influence of an external magnetic field on the electrons of a crystal. In the late 1970s, numerical studies for the most prominent model, the almost Mathieu operator (AMO), produced the first example of a fractal in physics known as "Hofstadter's butterfly," marking the starting point for the on-going strong interest in such operators in both mathematics and physics. Whereas research in the first three decades was focussed mainly on unraveling the unusual properties of the AMO, in recent years a combination of techniques from dynamical systems with those from spectral theory has allowed for a more "global," model-independent point of view. Intriguing phenomena first encountered for the AMO, notably the appearance of a critical phase corresponding to purely singular continuous spectrum, could be tested for their prevalence in general models. This workshop will introduce the participants to some of the techniques necessary to understand the spectral properties of quasi-periodic Schrödinger operators. The presentation is of expository nature and will particularly emphasize the close ties to dynamical systems (``matrix cocycles''), which was successfully used to address several open problems (e.g. the ``Ten Martini problem'') and enabled above-mentioned global perspective.

Topics included are: Basics: collectivity and regularity of spectral properties, matrix cocycles, arithmetic conditions Lyapunov exponent: positivity and continuity Supercritical behavior - Anderson localization Subcritical behavior - Duality, reducibility, and absolutely continuous spectrum Avila's global theory and critical behavior

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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons