我们调查的定量（或有效）版本的限制吸收原理，薛定谔运营商的渐近锥流形与短程潜力，特别是考虑估计的形式$$ \| R(λ+i\eps) f \|_{H^{0,-1/2-σ}} \leq C(λ, H) \| f \|_{H^{0,1/2+σ}}.$$我们特别感兴趣的确切性质的依赖性常数$C(λ,H)$对$λ$和$H$。事实证明，这个问题的答案是相当微妙的，在低能量$λ\ll 1$、中能量$λ\sim 1$和大能量$λ\gg 1$之间进行区分，并且在单个算子$H$的“定性”估计之间也存在非平凡的区别（可能服从某些谱条件，例如非共振，或几何条件，例如非俘获），和“定量”估计（对于某类中的所有算子$H$一致成立）。利用初等方法（分部积分法和常微分方程技术），我们对这些问题给出了一些尖锐的答案。作为这些估计的应用，在对算子$H$的一些附加假设下，我们给出了相关含时薛定谔方程的整体时间局部光滑估计和逐点衰减估计，以及相应波动方程解的积分局部能量衰减估计和逐点衰减估计.
英文摘要：
We investigate quantitative (or effective) versions of the limiting absorption principle, for the Schrödinger operator on asymptotically conic manifolds with short-range potentials, and in particular consider estimates of the form $$ \| R(λ+i\eps) f \|_{H^{0,-1/2-σ}} \leq C(λ, H) \| f \|_{H^{0,1/2+σ}}.$$ We are particularly interested in the exact nature of the dependence of the constants $C(λ,H)$ on both $λ$ and $H$. It turns out that the answer to this question is quite subtle, with distinctions being made between low energies $λ\ll 1$, medium energies $λ\sim 1$, and large energies $λ\gg 1$, and there is also a non-trivial distinction between "qualitative" estimates on a single operator $H$ (possibly obeying some spectral condition such as non-resonance, or a geometric condition such as non-trapping), and "quantitative" estimates (which hold uniformly for all operators $H$ in a certain class). Using elementary methods (integration by parts and ODE techniques), we give some sharp answers to these questions. As applications of these estimates, we present a global-in-time local smoothing estimate and pointwise decay estimates for the associated time-dependent Schrödinger equation, as well as an integrated local energy decay estimate and pointwise decay estimates for solutions of the corresponding wave equation, under some additional assumptions on the operator $H$.
An annulus multiplier and applications to the Limiting absorption principle for Helmholtz equations with a step potential
我们考虑Helmholtz方程$-Δu+V \, u - λ\, u = f $在$\mathbb{R}^n$上，其中势$V:\mathbb{R}^n\to\mathbb{R}$在每个半空间$\mathbb{R}^{n-1}\times (-\infty,0)$和$\mathbb{R}^{n-1}\times (0,\infty)$上都是常数.本文利用Fourier限制理论证明了一个$L^p-L^q$ --频率为$λ>\max \, V$的限制吸收原理，并导出了线性和非线性Helmholtz方程非平凡解的存在性。
Limiting absorption principle for time-harmonic elastic scattering of plane waves from diffraction gratings
Three Gorges Mathematical Research Center, College of Mathematics and Physics, China Three Gorges University（三峡大学数学与物理学院）；School of Mathematical Sciences and LPMC, Nankai University（南开大学数学科学学院）
Wave propagation in the frequency regime in one-dimensional quasiperiodic media -Limiting absorption principle
Columbia University（哥伦比亚大学）；APAM, Columbia University, United States（美国哥伦比亚大学应用物理与应用数学系）；POEMS, CNRS, Inria, ENSTA, Institut Polytechnique de Paris,91120 Palaiseau, France（法国帕莱索欧洲光学与测量工程实验室（POEMS），法国国家科学研究中心（CNRS），法国国家信息与自动化研究所（Inria），国立高等先进技术学院（ENSTA），巴黎理工学院）
Spectral theory for Maxwell's equations at the interface of a metamaterial. Part II: Limiting absorption, limiting amplitude principles and interface resonance
The properties of the solution for the Cauchy problem of a double nonlinear time-dependent parabolic equation in non-divergence form with a source or absorption
