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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 ĈS(δ) denote this optimal prefactor. Assuming that the uncontrolled region contains a nonempty open set and that the observability cost satisfies C(T)≤ C0eC0/T for small T>0, we prove ec√δ≤ ĈS(δ)≤ eC√δ, δ≫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)≲ eC/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 ℝn with exterior-ball control; and bounded smooth domains satisfying the geometric control condition for generalized geodesics.

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Cite

BibTeXRIS
BibTeX
@misc{huang2026asymptotic,
  title = {{Asymptotic Growth of Optimal Stabilization Prefactors for the Schrödinger Equations}},
  author = {Shanlin Huang and Changqin Quan and Gengsheng Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.28055},
  eprint = {2609.28055},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.28055},
}
RIS
TY  - UNPB
TI  - Asymptotic Growth of Optimal Stabilization Prefactors for the Schrödinger Equations
AU  - Shanlin Huang
AU  - Changqin Quan
AU  - Gengsheng Wang
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.28055
UR  - https://arxiv.org/abs/2609.28055
AB  - 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 Ĉ_S(δ) denote this optimal prefactor. Assuming that the uncontrolled region contains a nonempty open set and that the observability cost satisfies C(T)≤ C_0e^C_0/T for small T>0, we prove e^c√δ≤ Ĉ_S(δ)≤ e^C√δ, δ≫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)≲ 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 ℝ^n with exterior-ball control; and bounded smooth domains satisfying the geometric control condition for generalized geodesics.
ER  -