MIT PDE/Analysis Seminar

Tuesday, February 27, 2024 at 4:00pm to 5:30pm

MIT, 2-136 182 Memorial Drive, Cambridge, MA

Speaker:  Antoine Gloria (Sorbonne Université and Universitè Libre de Bruxelles)

Title:  Large-scale dispersive estimates for acoustic operators: homogenization meets localization

Abstract:  At low frequencies the acoustic operator with random coefficients essentially behaves like a Laplacian (the so-called homogenized operator). We might thus expect the associated wave operator to display some dispersion.  By blending standard dispersive estimates for homogenized operators and quantitative homogenization of the wave equation, we derive some ``weak’’ (say, large-scale) dispersive estimates for waves in disordered media.  Applied to the spreading of low-energy eigenstates, they allow us to relate quantitatively homogenization to Anderson localization for acoustic operators in disordered media. This gives a short and direct proof that the lower spectrum of the acoustic operator is purely absolutely continuous in case of periodic media, and it further provides new lower bounds on the localization length of possible eigenstates in case of quasiperiodic or random media.

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