Propagation of singularities and Fredholm analysis for the time-dependent Schr\"odinger equation

We study the time-dependent Schr\"odinger operator $P = D_t + \Delta_g + V$ acting on functions defined on $\mathbb{R}^{n+1}$, where, using coordinates $z \in \mathbb{R}^n$ and $t \in \mathbb{R}$, $D_t$ denotes $-i \partial_t$, $\Delta_g$ is the positive Laplacian with respect to a time dependent family of non-trapping metrics $g_{ij}(z, t) dz^i dz^j$ on $\mathbb{R}^n$ which is equal to the Euclidean metric outside of a compact set in spacetime, and $V = V(z, t)$ is a potential function which is also compactly supported in spacetime. In this paper we introduce a new approach to studying $P$, by finding pairs of Hilbert spaces between which the operator acts invertibly. Using this invertibility it is straightforward to solve the `final state problem' for the time-dependent Schr\"odinger equation, that is, find a global solution $u(z, t)$ of $Pu = 0$ having prescribed asymptotics as $t \to \infty$. These asymptotics are of the form $$ u(z, t) \sim t^{-n/2} e^{i|z|^2/4t} f_+\big( \frac{z}{2t} \big), \quad t \to +\infty $$ where $f_+$, the `final state' or outgoing data, is an arbitrary element of a suitable function space $\mathcal{W}^k(\mathbb{R}^n)$; here $k$ is a regularity parameter simultaneously measuring smoothness and decay at infinity. We can of course equally well prescribe asymptotics as $t \to -\infty$; this leads to incoming data $f_-$. We consider the `Poisson operators' $\mathcal{P}_\pm : f_\pm \to u$ and precisely characterize the range of these operators on $\mathcal{W}^k(\mathbb{R}^n)$ spaces. Finally we show that the scattering matrix, mapping $f_-$ to $f_+$, preserves these spaces.