Characterizations of stabilizable sets for some parabolic equations in ${\mathbb{R}}^n$

We consider the parabolic type equation in ${\mathbb{R}}^n$: \begin{align}\label{equ-0} (\partial_t+H)y(t,x)=0,,,, (t,x)\in (0,\infty)\times$\mathbb{R}$^n;;; \quad y(0,x)\in L^2($\mathbb{R}$^n), \end{align} where $H$ can be one of the following operators: (i) a shifted fractional Laplacian; (ii) a shifted Hermite operator; (iii) the Schr"{o}dinger operator with some general potentials. We call a subset $E\subset {\mathbb{R}}^n$ as a stabilizable set for the above equation, if there is a linear bounded operator $K$ on $L^2({\mathbb{R}}^n)$ so that the semigroup $\{e^{-t(H-{\chi }_{E}K)}{\}}_{t\ge 0}$ is exponentially stable. (Here, ${\chi }_E$ denotes the characteristic function of $E$, which is treated as a linear operator on $L^2({\mathbb{R}}^n)$.) This paper presents different geometric characterizations of the stabilizable sets for the above equation with different $H$. In particular, when $H$ is a shifted fractional Laplacian, $E\subset {\mathbb{R}}^n$ is a stabilizable set if and only if $E\subset {\mathbb{R}}^n$ is a thick set, while when $H$ is a shifted Hermite operator, $E\subset {\mathbb{R}}^n$ is a stabilizable set for if and only if $E\subset {\mathbb{R}}^n$ is a set of positive measure. Our results, together with the results on the observable sets for the above equation obtained in \cite{AB,Ko,Li,M09}, reveal such phenomena: for some $H$, the class of stabilizable sets contains strictly the class of observable sets, while for some other $H$, the classes of stabilizable sets and observable sets coincide. Besides, this paper gives some sufficient conditions on the stabilizable sets for the above equation where $H$ is the Schr"{o}dinger operator with some general potentials.