Observable sets, potentials and Schrödinger equations

Abstract

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 $\gamma >0$ and $L>0$ so that $$ \left|E \bigcap [x, x+ L]\right|\geq γL\;\;\mbox{for each}\;\;x\in \mathbb{R}; $$ Second, when $m=1$ ($m\ge 2$ resp.), $E$ is an observable set at some time (at any time resp. ) if and only if it is weakly thick, namely $$\underset{x\to +\infty }{\underline{\lim }}\frac{|E\bigcap [-x,x]|}{x}>0.$$ 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 $T>\frac{\pi }{2}$.

50 pages