The Analysis and PDE Seminar will take place on Monday, September 26, in room 740, Evans Hall, from 4:10-5:00 pm.

Speaker: Satoshi Masaki

Title: Minimization problems on non-scattering solutions to NLS equation

Abstract: We consider global dynamics of focusing nonlinear Schrodinger equations. A first step in this direction is small data scattering which tells us that solutions around the zero solution asymptotically behave like free solutions. On the other hand, there exists non-scattering solutions such as standing waves and blowing-up solutions.

In this talk, we will seek threshold solutions between scattering solutions around zero and solutions with other behaviors, by introducing two minimization problems on non-scattering solutions. In particular, our main interest is the analysis of mass-subcritical case, in which the ground states are stable. The analysis of the minimization problems are based on concentration compactness/rigidity argument initiated by Kenig and Merle.

The Analysis and PDE Seminar will take place on Monday, September 12, in room 740, Evans Hall, from 4:10-5:00 pm.

Speaker: Semyon Dyatlov

Title: Resonances for open quantum maps

Abstract: Quantum maps are a popular model in physics: Symplectic relations on tori are quantized to produce families of $N\times N$ matrices and the high energy limit corresponds to the large $N$ limit. They share a lot of features with more complicated quantum systems but are easier to study numerically. We consider open quantum baker’s maps, whose underlying classical systems have a hole allowing energy escape. The eigenvalues of the resulting matrices lie inside the unit disk and are a model for scattering resonances of more general chaotic quantum systems. However in the setting of quantum maps we obtain results which are far beyond what is known in scattering theory.

We establish a spectral gap (that is, the spectral radius of the matrix is separated from 1 as $N\to\infty$) for all the systems considered. The proof relies on the notion of fractal uncertainty principle and uses the fine structure of the trapped sets, which in our case are given by Cantor sets, together with simple tools from harmonic analysis, algebra, combinatorics, and number theory. We also obtain a fractal Weyl upper bound for the number of eigenvalues in annuli. These results are illustrated by numerical experiments which also suggest some conjectures.

This talk is based on joint work with Long Jin.