This event will only take place online. Contact [email protected] for the Zoom link.

Speaker: Antoine Gloria, Sorbonne Université

Title: "Approximate Floquet Bloch theory for waves in quasi-periodic media"

Abstract: In this talk I will introduce some approximate variant of the Floquet-Bloch theory to study wave propogation in quasi-periodic media in the small coupling regime. Floquet-Bloch theory allows one to diagonalise periodic operators, and extended states are given by periodic modulations of plane waves (called Bloch waves). Although it might be natural to look for extended states of quasi-periodic operators in form of quasi-periodic modulations of plane waves, we rapidly face a lack of compactness which prevents to prove their existence by soft arguments. The starting point of our work is the observation that, in the periodic setting, the derivatives of Bloch waves with respect to the wave number solve a PDE with periodic coefficients. Perturbation theory even ensures the summability of the Taylor series. In the quasi-periodic setting, whereas Bloch waves might not exist, these PDEs are also well-posed, and we can consider the corresponding partial sums. By construction these partial sums approximately diagonalise the quasi-periodic operator, up to an error that can be quantified. A careful analysis of this error together with weighted energy estimates on the Schrödinger equation allow us to prove long (although not infinite) time ballistic transport for small coupling constants. The approach is reminiscent of normal forms and Nehorosev stability.

This is based on joint works with Antoine Benoit, Mitia Duerinckx, and Christopher Shirley.

Brief biography: Antoine Gloria is a professor at Sorbonne Université. His field of expertise is PDEs in random media.