Question : For what condition on $V$ (we can take it smooth, bounded, whatever necessary), the one-parameter unitary group $U(t)$ associated to the seladjoint operator $A=-\Delta+V$ on $\mathbb{R}^n$ preserves the space of Schwartz (rapidly decaying) functions.
EDIT : the following questions here have been answered negatively:
The following fact is already known :
Let $A$ be a selfadjoint operator on a Hilbert space $H$, and let $D$ be a dense subset of $H$, that is included in the domain of $A$. We denote $U(t)$ the one-parameter group of unitary operators associated to $A$. Then :
If $U(t) D \subseteq D$ for all $t$ in an open interval containing $0$, then $D$ is a core for $A$. (i.e. $A$ is essentially selfadjoint on $D$).
1) Does the inverse implication hold ? That is : if $D$ is a core for $A$, is it true that $U(t)$ preserves $D$ ? (for all $t$ in an interval containing $0$)
2) If the answer is no, my question is then the same but for specific example of $A$ and $H$ : do we have the inverse implication when $H=L^2(\mathbb{R}^n)$ and $A$ is a Schrödinger operator $-\Delta+ V$, provided $V$ has "good" properties ?