Fredholm approach to the Schrödinger equation
Speaker: Jesse Gell-Redman
Zoom: 293 812 9202 Passcode: BIMSA
We discuss a new approach, inspired by work of Hintz and Vasy, to solving the Schrödinger equation $(i \partial_t - \Delta) u = f$ using the Fredholm method. Specifically, we use 'parabolic' pseudodifferential operators (reflecting the parabolic nature of the symbol of $P = i \partial_t - \Delta$) to obtain families of function spaces $X, Y$ for which $P : X \to Y$ is an isomorphism. The spaces further allow us to read off precise regularity and decay information about $u$ directly from that of $f$. We discuss applications to the nonlinear Schrödinger equation, and extensions of this method to equations with compact spatial perturbations, such as smooth decaying potential functions, using the N-body calculus of Vasy. This includes joint work with Dean Baskin, Sean Gomes, and Andrew Hassell.
[Seminar on microlocal analysis and applications]