初态到终态逆问题的稳定性估计
Stability estimates for the initial-to-final-state inverse problem
- The Johann Radon Institute for Computational and Applied Mathematics（约翰·拉东计算与应用数学研究所）
- Aarhus University（奥胡斯大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对薛定谔方程初态到终态逆问题，量化了其唯一性结果，证明随时间变化电势的稳定性为对数型，与时间无关电势的稳定性为赫尔德型。
AI中文摘要：
薛定谔方程的初态到终态逆问题，是在已知初态到终态映射的前提下，唯一确定生成演化的哈密顿量。该泛函将每个初态 $f\in L^2(\mathbb{R}^n)$ 映射为固定时刻 $T$ 对应的终态。该问题由Caro和Ruiz针对来自随时间变化、在无穷远处呈超指数衰减的有界电势的哈密顿量提出；Caro、Parissis与本文作者证明，对于在无穷远处呈超线性衰减的与时间无关的有界电势，唯一性也成立。本文量化上述唯一性结果，证明电势在初态到终态映射的微小变化下是稳定的：对于随时间变化的电势，得到对数型稳定性；对于与时间无关的电势，取得显著改进，证明其满足赫尔德稳定性估计。
英文摘要：
The initial-to-final-state inverse problem for the Schrödinger equation consists in determining uniquely the Hamiltonian that generates the evolution, assuming the knowledge of the initial-to-final-state map. This functional maps each initial state $f\in L^2(\mathbb{R}^n)$ to the corresponding final state at a fixed time $T$. The problem was formulated by Caro and Ruiz in the case of Hamiltonians arising from time-dependent bounded electric potentials that exhibit super-exponential decay at infinity. Caro, Parissis and the authors of this article established that uniqueness also holds for time-independent bounded potentials with super-linear decay at infinity. In this paper, we quantify the above uniqueness results establishing that the potentials are stable under small changes of the initial-to-final-state maps. In the case of time-dependent potentials we get a stability of logarithmic type. A notable improvement is achieved when the potentials are time-independent, where under this assumption we prove Hölder stability estimates.
