论文标题
特征性区域的征函数限制的下限
Lower bounds for eigenfunction restrictions in lacunary regions
论文作者
论文摘要
令$（m，g）$为紧凑，光滑的riemannian歧管，$ \ {u_h \} $为$ l^2 $ normalization laplace eigenFunctions的序列，在$ m \ setMinus \ supt $ fext {supp {supp} $ neq中，具有局部缺陷量$μ$ $π：t^*m \ to m $是规范投影。使用Carleman估计，我们证明，对于任何真正平滑的封闭的高度张开$ h \ subset（m \ setMinus \ text {supp}（supp}（π_*μ））$足够接近$ \ text {supp} {supp}（π_*μ），$，$ ge__ $ ge__________ e^{ - [\，d（h，\ text {supp}（π_*μ）） + \，δ] /h} $$ as $ h \ to 0^ + $。我们还表明，结果适用于Schrödinger操作员的本征函数，并将完全集成（QCI）系统的扭曲产品和联合特征功能应用于特征函数。
Let $(M,g)$ be a compact, smooth Riemannian manifold and $\{u_h\}$ be a sequence of $L^2$-normalized Laplace eigenfunctions that has a localized defect measure $μ$ in the sense that $ M \setminus \text{supp}(π_* μ) \neq \emptyset$ where $π:T^*M \to M$ is the canonical projection. Using Carleman estimates we prove that for any real-smooth closed hypersurface $H \subset (M\setminus \text{supp} (π_* μ))$ sufficiently close to $ \text{supp}(π_* μ),$ and for all $δ>0,$ $$ \int_{H} |u_h|^2 dσ\geq C_δ\, e^{- [\, d(H, \text{supp}(π_* μ)) + \,δ] /h} $$ as $h \to 0^+$. We also show that the result holds for eigenfunctions of Schrödinger operators and give applications to eigenfunctions on warped products and joint eigenfunctions of quantum completely integrable (QCI) systems.
