Analysis/PDE Reading Seminar

Date: October 31, 2017

Time: 4:00PM - 5:00PM

Location: BLOC 624

Speaker: Dean Baskin, TAMU

Title: A brief introduction to resonances

Abstract: Spectral theory for the Laplacian on compact manifolds gives you a discrete set of eigenvalues and an orthonormal basis of eigenfunctions. For non-compact problems, there is typically continuous spectrum and only finitely many eigenvalues. Is there anything resembling a discrete family of eigenvalues in this context? In some sense, the answer is yes: these are the resonances. In this talk I will provide a brief introduction to the theory of resonances. I will provide some motivation for their study by working through the case of the one-dimensional wave equation. I will then talk about resonances on Euclidean and hyperbolic spaces, where they can be calculated explicitly. Finally I will provide some discussion of how to define them in the case of potential scattering (and maybe some geometric scattering) via the analytic Fredholm theorem. In a future talk I will use the explicit calculation of the resonances on hyperbolic space to calculate the asymptotic behavior of the wave equation on Minkowski space.