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

Abstract: We study the time-dependent Schr\"odinger operator P=Dₜ+g+V acting on functions defined on R^n+1, where, using coordinates zⁿ and t, Dₜ denotes -iₜ, g is the positive Laplacian with respect to a time dependent family of nontrapping metrics g₈₉ (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 e^i|z|²/4tf_+ (z2t), 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.