PROPAGATION ESTIMATES AND FREDHOLM ANALYSIS FOR THE TIME-DEPENDENT SCHRÖDINGER EQUATION

Abstract

We study the time-dependent Schrödinger operator P = Dt + ∆g + V acting on functions defined on (Formula presented), where, using coordinates (Formula presented) and t ∈ ℝ, Dt denotes −i∂t, ∆g is the positive Laplacian with respect to a time dependent family of nontrapping metrics gij(z,t)dzidzj 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..