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Open PDE & Analysis Seminar

Prof Svitlana Mayboroda (University of Minnesota )

The landscape law for the integrated density of states

Friday 19th June 2020, 4pm (Paris time)

Abstract : Complexity of the geometry, randomness of the potential, and many other irregularities of the system can cause powerful, albeit quite different, manifestations of localization, a phenomenon of sudden confinement of waves, or eigenfunctions, to a small portion of the original domain. In the present talk we show that behind a possibly disordered system there exists a structure, referred to as a landscape function, which can predict the location and shape of the localized eigenfunctions, a pattern of their exponential decay, and deliver accurate bounds for the corresponding eigenvalues. In particular, we establish the "landscape law", the first non-asymptotic estimates from above and below on the integrated density of states of the Schroedinger operator using a counting function for the minima of the localization landscape. The results are deterministic, and rely on a new uncertainty principle. Narrowing down to the context of disordered potentials, we derive the best currently available bounds on the integrated density of states for the Anderson model.

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Thomas Alazard (ENS Paris-Saclay)         Nicolas Burq (Orsay)         Iván Moyano (Nice)