András Vasy (Stanford)

The Analysis and PDE seminar will take place Monday, February 5, in 740 Evans from 4:10 to 5 pm.

Title: Fredholm theory and the resolvent of the Laplacian near zero energy on asymptotically conic spaces

Abstract: We consider geometric generalizations of Euclidean low energy resolvent estimates, such as estimates for the resolvent of the Euclidean Laplacian plus a decaying potential, in a Fredholm framework. More precisely, the setting is that of perturbations $P(\sigma)$ of the spectral family of the Laplacian $\Delta_g-\sigma^2$ on asymptotically conic spaces $(X,g)$ of dimension at least $3$, and the main result is uniform estimates for $P(\sigma)^{-1}$ as $\sigma\to 0$ on microlocal variable order spaces under an assumption on the nullspace of $P(0)$ on the appropriate function space (which in the Euclidean case translates to $0$ not being an $L^2$-eigenvalue or having a half-bound state). These spaces capture the limiting absorption principle for $\sigma\neq 0$ in a lossless, in terms of decay, manner.