Invariance of sets under Schrödinger equations

Question

We are considering the Schrödinger equation on $\mathbb{R}^d \times [0,T]$

$$i \partial_t \psi(x,t)=-\Delta \psi(x,t) + u(t)V(x) \psi(x,t), t>0$$ $$\psi(x,0):=\psi(x_0) \in L^2(\mathbb{R}^d)$$ with $u\in C([0,T])$ and $V \in C_c(\mathbb{R}^d).$

Standard picard iteration shows that this equation has a unique solution.

Are there any sufficient conditions that the flow of this evolution equation leaves closed convex sets invariant?

I could for example imagine that if $X$ is such a set and $\psi(x_0)\in X$ and for all $y \in X$ and $t>0$ $$\langle \partial_t\psi(x,t), y \rangle =0$$ that this condition is true. I do not know if this is indeed true, but this was a possible condition that came first to my mind.

Answer 1

If $uV$ is real valued, then the convex set of complex functions $\psi$ with $\|\psi\|_{L^2}\le A$ ($A$ a given constant) is invariant under the flow.

It is hard to find something else, because the Schrödinger equation tends to transform properties of the initial data into something different. See dispersion estimates of Strichartz estimates. For instance, if the initial data is compactly supported, then the solution is smooth for $t>0$.