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arXiv · 2609.28055

Asymptotic Growth of Optimal Stabilization Prefactors for the Schrödinger Equations

Shanlin Huang·Changqin Quan·Gengsheng Wang
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Abstract

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 $δ>0$, let $\widehat{C}_S(δ)$ denote this optimal prefactor. Assuming that the uncontrolled region contains a nonempty open set and that the observability cost satisfies $C(T)\le C_0e^{C_0/T}$ for small $T>0$, we prove $$ e^{c\sqrtδ}\le \widehat{C}_S(δ)\le e^{C\sqrtδ}, \qquad δ\gg1. $$ 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 $C(T)\lesssim e^{C/T}$. 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.

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Shanlin Huang, Changqin Quan, Gengsheng Wang. 2026-09-23. Asymptotic Growth of Optimal Stabilization Prefactors for the Schrödinger Equations. https://arxiv.org/abs/2609.28055

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