Time: Monday, 4pm.
First, I will discuss the asymptotic behavior of the corresponding Krein spectral shift function (SSF) near the Landau levels which play the role of spectral thresholds. I will show that the SSF has singularities near these thresholds, which admit a fairly explicit description in terms of appropriate Berezin-Toeplitz operators.
Further, I will introduce the resonances of the perturbed Schroedinger operator, and will discuss their asymptotic distribution near the spectral thresholds. I will show that under suitable assumptions on the pertubing potential, there are infinitely many resonances near every fixed Landau level, and the main asymptotic term of the corresponding resonance counting function is again related to the Berezin-Toeplitz operators arising in the description of the SSF singularities.
If time allows, some extensions to Pauli and Dirac operators with non-constant magnetic fields could be briefly outlined.
Kerr-de Sitter black holes are rotating black holes in a universe with a positive cosmological constant, given by an explicit formula, due to Kerr and Carter. They are parameterized by their mass and angular momentum, much like Kerr black holes in a space-time with vanishing cosmological constant.
I will discuss analytic (PDE) aspects of the stability question for these black holes in the initial value formulation. Namely, appropriately interpreted (fixing a gauge), Einstein's equations can be thought of as quasilinear wave equations, and then the question is if perturbations of the initial data along a Cauchy hypersurface produce solutions which are close to, and indeed asymptotic to, a Kerr-de Sitter black hole, typically with a different mass and angular momentum. The key aspects are arranging appropriate gauge conditions and the analysis of linear wave equations on asymptotically Kerr-de Sitter backgrounds, including with coefficient possessing limited regularity.
We establish a spectral gap (that is, the spectral radius of the matrix is separated from 1 as N tends to infinity 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.