Observable Sets, Potentials and Schrödinger Equations

Consider the Schrödinger equation: $$\mathrm{i}\partial _tu=Hu$$ over $$\mathbb {R}^n$$ , where H is a self-adjoint operator on $$L^2(\mathbb {R}^n)$$ which sum of $$-\Delta $$ and some potential. This paper aims to study observability for above equation, including observable sets time. We mention that measurable subset $$E\subset \mathbb called an set at time $$T>0$$ if there constant $$C>0$$ (depending T E) such $$\begin{aligned} \int _{\mathbb {R}^n}{|u_0(x)|^2\,\mathrm {d}x}\le C\int _0^{T}\int _E{|e^{-\mathrm{i}tH}u_0|^2\,\mathrm {d}x}\,\mathrm {d}t\; \text{ } all u_0\in L^2(\mathbb {R}^n). \end{aligned}$$ First, we characterize 1-dim case $$H=-\partial _x^2+x^{2m}$$ (with $$m\in {N}:=\{0,1,\dots \}$$ ). More precisely, obtain what follows: (i) When $$m=0$$ {R}$$ only it thick, namely, are constants $$\gamma ,L>0$$ so \left| E \bigcap [x, x+ L]\right| \ge \gamma L\;\;\text{ each }\;\;x\in {R}; (ii) $$m=1$$ ( $$m\ge 2$$ resp.), (at any resp.) weakly namely \varliminf _{x \rightarrow +\infty \frac{|E\bigcap [-x, x]|}{x} >0. These reveal how potentials $$x^{2m}$$ affect observability. Second, follows n-dim $$H=-\Delta +|x|^2$$ (the Harmonic oscillator): For $$r>0$$ exterior domain $$B^c(0,\, r)$$ time; Let $$E_1$$ be half bisected by hyperplane across origin. Then $$T>\frac{\pi }{2}$$ .