Observable sets, potentials and Schr\"{o}dinger equations

We characterize observable sets for 1-dim Schr\"{o}dinger equations in $\mathbb{R}$: $i \partial_t u = (-\partial_x^2+x^{2m})u$ (with $m\in \mathbb{N}:=\{0,1,\dots\}$). More precisely, we obtain what follows: First, when $m=0$, $E\subset\mathbb{R}$ is an observable set at some time if and only if it is thick, namely, there is $\gamma>0$ and $L>0$ so that $$\left|E \bigcap [x, x+ L]\right|\geq \gamma L\;\;\mbox{for each}\;\;x\in \mathbb{R};$$ Second, when $m=1$ ($m\geq 2$ resp.), $E$ is an observable set at some time (at any time resp. ) if and only if it is weakly thick, namely $$\varliminf_{x \rightarrow +\infty} \frac{|E\bigcap [-x, x]|}{x} >0.$$ From these, we see how potentials $x^{2m}$ affect the observability (including the geometric structures of observable sets and the minimal observable time). Besides, we obtain several supplemental theorems for the above results, in particular, we find that a half line is an observable set at time $T>0$ for the above equation with $m=1$ if and only if $T>\frac{\pi}{2}$.