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

We study the time-dependent Schr\"odinger operator P=Dₜ+Δg+V acting on functions defined on Rⁿ⁺¹, where, using coordinates zⁿ and t, Dₜ denotes -i∂ₜ, Δg is the positive Laplacian with respect to a time dependent family of nontrapping metrics gᵢⱼ(z,t)dzⁱ dzʲ on Rⁿ 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→ +∞. These asymptotics are of the form \[ u(z,t)~ t-n/2 ei|z|^2/4tf_+(z/2t), t→ +∞ \] where f₊, the `final state' or outgoing data, is an arbitrary element of a suitable function space Wᵏ(Rⁿ); here k is a regularity parameter simultaneously measuring smoothness and decay at infinity. We can of course equally well prescribe asymptotics as t→ -∞; this leads to incoming data f₋. We consider the `Poisson operators' P_± : f_±→ u and precisely characterise the range of these operators on Wᵏ(Rⁿ) spaces. Finally we show that the scattering map, mapping f₋ to f₊, preserves these spaces.