Asymptotic Growth of Optimal Stabilization Prefactors for the Schrödinger Equations

We study the asymptotic behavior of the optimal stabilization prefactor for the free Schrödinger equation, optimized over all bounded stationary linear feedbacks. For a prescribed decay rate $\delta >0$, let ${\widehat{C}}_S(\delta )$ denote this optimal prefactor. Assuming that the uncontrolled region contains a nonempty open set and that the observability cost satisfies $C(T)\le C_0{e}^{C_0\mathrm{/}T}$ for small $T>0$, we prove $$e^{c\sqrt{\delta }}\le {\widehat{C}}_S(\delta )\le e^{C\sqrt{\delta }},\delta \gg 1.$$ The lower bound is based on the uncontrolled open hole and a quantitative cutoff construction adapted to the Laplacian, while the upper bound combines an exponentially weighted Gramian with the small-time observability cost $C(T)\lesssim e^{C\mathrm{/}T}$. The results apply to controlled Schrödinger equations in three settings: flat tori, including a strip-control example beyond the geometric control condition; the whole space ${\mathbb{R}}^n$ with exterior-ball control; and bounded smooth domains satisfying the geometric control condition for generalized geodesics.