2022/01/10 by Jesse Gell‐Redman, Gell-Redman, Jesse, Sean Gomes +3 · 1 citation
Mathematics · #Advanced Mathematical Physics Problems #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2201.03140
We study the time-dependent Schrödinger operator P = Dt + Δg + V acting on functions defined on ℝn+1, where, using coordinates z ∈ ℝn and t ∈ ℝ, Dt denotes -i ∂t, Δg is the positive Laplacian with respect to a time dependent family of non-trapping metrics gij(z, t) dzi dzj on ℝn 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 → ∞. These asymptotics are of the form u(z, t) ∼ t-n/2 ei|z|2/4t f+( (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 characterize the range of these operators on Wk(ℝn) spaces. Finally we show that the scattering matrix, mapping f- to f+, preserves these spaces.