Propagation of singularities and Fredholm analysis for the time-dependent Schrödinger equation

Abstract

We study the time-dependent Schrödinger 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 $g_{ij}(z,t)d{z}^{i}d{z}^j$ 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 Schrödinger 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 $$u(z,t)\sim t^{-n/2}{e}^{i|z{|}^2/4t}{f}_{+}\left(\frac{z}{2t}\right),t\to +\infty$$ 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.

arXiv comment63 pages, 3 figures