论文标题
可观察的集合，电势和Schrödinger方程
Observable sets, potentials and Schrödinger equations
论文作者
论文摘要
我们表征了$ \ mathbb {r} $中的1-DimSchrödinger方程的可观察集，$ i \ partial_t u =（ - \ partial_x^2+x^{2m}）更确切地说，我们获得以下内容：首先，当$ m = 0 $，$ e \ subset \ mathbb {r} $是一个可观察到的设置，并且仅在且仅当它厚时，即$γ> 0 $和$ l> 0 $，$ l> 0 $，这样的$ $ \ weft | e \ e \ e \ bigCap [x，x+ l]每个} \; \; x \ in \ mathbb {r}; $$ second，当$ m = 1 $（$ m \ geq 2 $ resp。）时，$ e $在某个时候（任何时候）是一个可观察的设置，并且仅当且仅当它较弱的情况下，即$ \ varliminf_ {x \ rightArrow +rightArrow +\ rightarrow +\ frac}从这些$$中，我们看到了潜力$ x^{2M} $如何影响可观察性（包括可观察的集合的几何结构和最小的可观察时间）。 此外，我们为上述结果获得了几个补充定理，尤其是，我们发现半线是一个可观察到的时间$ t> 0 $的上述方程式，$ m = 1 $时，仅当$ t> \fracπ{2} $。
We characterize observable sets for 1-dim Schrödinger 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 $γ>0$ and $L>0$ so that $$ \left|E \bigcap [x, x+ L]\right|\geq γL\;\;\mbox{for each}\;\;x\in \mathbb{R}; $$ 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 $T>\fracπ{2}$.
