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

Andras Vasy (Stanford University)
Friday, October 26, 2018 - 2:30pm to 3:30pm
Padelford C-36


In this talk I will discuss 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; this is different from even the usual treatment of the Euclidean problem. 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 Sobolev 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.


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