Localization of solutions for time-dependent Schroedinger equation

Question

I've been playing around with numerical solutions to the Schroedinger equation and I came across something that feels very natural, but I was not able to prove it with the math I know.

The motivation comes from the following physical situation. Consider scattering of a wave-packet over a square potential barrier. If the wave-packet is far enough from the barrier it looks like it evolves according to the free equation, like there is no barrier at all (you can find a video in the post here). I would like to understand how one could see this mathematically. So:

1) What does it mean in this context for a $L^2$-function $\psi$ to be localized somewhere? It is clear that just taking the average $<x>= (\psi,x\psi)$ is not a good choice (consider, for example, a superposition of two Gaussian states). On the other hand simply saying that $\psi$ has compact support excludes all the Gaussians.

2) Given a potential $V(x) \in C^\infty_c(\mathbb{R}^n)$ and initial state $\psi_0(x)$ either localized in some sense away from the support of the potential or just in $C^\infty_c(\mathbb{R}^n)$ with $\text{supp } \phi \cap \text{supp } V = \emptyset $, is it true that the solution of the Schroedinger equation $$ i\partial_t \psi = (-\Delta+V(x))\psi, \quad \psi(0,x) = \psi_0(x) $$ is close to the corresponding solution of the free equation $$ i\partial_t \psi = -\Delta\psi, \quad \psi(0,x) = \psi_0(x) $$ for small times? I don't assume that $V$ is small or something, so I don't think that perturbation theory will be enough, since Schroedinger equation is not local.

Thank you for your help.

Answer 1

Here are some remarks to put Q2 into context, though I'm not answering the actual question. Basically, I'm going to give a quick summary of my 2007 paper Finite propagation speed and kernel estimates for Schrodinger operators; see reference 17 here.

If $H_1=-\Delta$ and $H_2=-\Delta+V$, and the support of $\psi$ is disjoint from the support of $V$, then the solutions of the wave equations $\psi_{tt}=H\psi$ will agree for a while since this equation has finite speed of propagation. In other words, $(\cos t\sqrt{H_j})\psi$ is independent of $j$ for a while.

This will imply the desired kind of statement for certain other functions $f(H_j)\psi$, by writing $f(\lambda)=\int g(x)\cos x\sqrt{\lambda}\, dx$ (essentially a Fourier transform if the spectra are positive), and if $g$ decays, then we can approximately cut off the integrals. This would work for example for the semigroups $e^{-tH_j}$, or in the discrete case, but for the actual operator $e^{-iH_j}$ we would need to work with the function $e^{-i\lambda^2}$, which doesn't have a decaying Fourier transform.