The PDE & Analysis seminar covers topics in PDE and analysis.

schedule Date & time
Date/time
30 Sep 2022 | 12 - 1pm
person Speaker

## Speakers

Andrew Hassell, ANU
next_week Event series

## Contact

contact_support Contact
Contact name
Rong Wang
Contact email

## Description

Abstract:

We consider at the time-dependent Schrodinger operator $P$ on $\mathbb{R}^{n+1}$, with fixed metric and potential that are flat/trivial outside a compact set in spacetime. Considering first the inhomogeneous equation

$Pu = f,$

in spacetime, we find Hilbert spaces of functions $P : X \to Y$ between which $P$ maps invertibly. This is done by proving microlocal propagation estimates, following Melrose and Vasy, near the characteristic variety of $P$, and assembling them into a global Fredholm estimate. Using this, we can solve the “final state problem”, which is to find a global solution to $Pu = 0$ where $u(x,t)$ has the asymptotic

$u(x, t) \sim t^{-n/2} e^{i|x|^2/4t} f_+(x/t), \quad t \to +\infty,$

for a prescribed function $f_+$.

Our framework leads to some precise results in linear scattering, which seem to be new. More significantly it suggests an entirely new approach to nonlinear scattering. I’ll mention a small data nonlinear scattering result which can be proved using this approach.

## Location

Seminar Room 1.33

Hanna Neumann Building 145