Propagation estimates and Fredholm analysis for the time-dependent Schrödinger equation

Abstract: 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 nontrapping metrics $g_{ij}(z,t)dz^i dz^j$ on $\mathbb{R}^n$ which are 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 characterise the range of these operators on $\mathcal{W}^k(\mathbb{R}^n)$ spaces. Finally we show that the scattering map, mapping $f_-$ to $f_+$, preserves these spaces.