Propagation of singularities and Fredholm analysis for the time-dependent Schr\"odinger equation

TL;DR

This paper introduces a new method for analyzing the time-dependent Schr"odinger equation by establishing invertibility between Hilbert spaces, enabling explicit solutions with prescribed asymptotics and detailed scattering analysis.

Contribution

The authors develop a novel approach to study the Schr"odinger operator using invertible Hilbert space pairs, allowing explicit construction of solutions with specified asymptotic behavior and characterization of scattering operators.

Findings

Constructed invertible pairs of Hilbert spaces for the Schr"odinger operator.

Solved the final state problem with prescribed asymptotics at infinity.

Characterized the range of Poisson operators and proved scattering matrix preserves function spaces.

Abstract

We study the time-dependent Schr\"odinger 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 $g_{ij}(z, t) dz^i dz^j$ 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 Schr\"odinger 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 $$ u(z, t) \sim t^{-n/2} e^{i|z|^2/4t} f_+\big( \frac{z}{2t} \big), \quad t \to +\infty $$ 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.