Time-dependent Schrodinger equation via microlocal analysis and Fredholm theory


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.