AI 中文总结
该研究针对带实值有界势的一维四阶分数拉普拉斯薛定谔方程，建立定量Landis估计，通过变换推导Grushin方程及谱估计，将局部势依赖转化为全局速率并传递非零性至边界。
AI 中文摘要
我们在实数域$\boldsymbol{\text{R}}$中，针对带有实值有界势的一维分数阶薛定谔方程$(-\boldsymbol{\text{Δ}})^{1/4}u+V(x)u=0$，建立了定量Landis估计。若$\boldsymbol{\text{V}}$的$L^\boldsymbol{\text{∞}}$范数$\boldsymbol{\text{≤}}1$，$u$的$L^\boldsymbol{\text{∞}}$范数$\boldsymbol{\text{≤}}C_0$，且$u$在区间$(-1,1)$上的$L^2$范数$\boldsymbol{\text{≥}}1$，则对所有足够大的$R$，有$\boldsymbol{\text{inf}}_{|x_0|=R}\boldsymbol{\text{‖}}u\boldsymbol{\text{‖}}_{L^\boldsymbol{\text{∞}}(x_0-1,x_0+1)}\boldsymbol{\text{≥}}\boldsymbol{\text{exp}}(-CR\boldsymbol{\text{log}}R)$。经过Caffarelli–Silvestre延拓和替换$y=z^2/2$，该方程退化为退化线上带有弱Robin条件的Grushin方程。对应的角算子经半密度共轭后具有算术谱$\boldsymbol{\text{κ}}_n=2n+\tfrac{1}{2}$。核心谱估计为：当参数与角晶格分离时，对所有$\boldsymbol{\text{ξ}}\boldsymbol{\text{∈}}\boldsymbol{\text{R}}$有$\boldsymbol{\text{sup}}_{\boldsymbol{\text{ξ}}\boldsymbol{\text{∈}}\boldsymbol{\text{R}}}\boldsymbol{\text{‖}}C((\boldsymbol{\text{τ}}+i\boldsymbol{\text{ξ}})^2-L_0)^{-1}C^*\boldsymbol{\text{‖}}\boldsymbol{\text{≤}}C\boldsymbol{\text{τ}}^{-1/2}$，它给出了吸收阈值$\boldsymbol{\text{τ}}\boldsymbol{\text{≥}}C(1+\boldsymbol{\text{‖}}V\boldsymbol{\text{‖}}_\boldsymbol{\text{∞}}^2)$下可测Robin反馈的线性权重Carleman估计。随后，定量内向传播、固定尺度Grushin球传播及内部柱插值估计将整体非零性传递到边界，Landis重标度将局部势依赖$C\boldsymbol{\text{‖}}q\boldsymbol{\text{‖}}_\boldsymbol{\text{∞}}^2$转化为全局速率$CR\boldsymbol{\text{log}}R$。
英文摘要
We establish a quantitative Landis estimate for the one-dimensional fractional Schrödinger equation $(-Δ)^{1/4}u+V(x)u=0$ in $\mathbb R$ with a real-valued bounded potential. If $\|V\|_{L^\infty}\le 1$, $\|u\|_{L^\infty}\le C_0$, and $\|u\|_{L^2(-1,1)}\ge 1$, then \[ \inf_{|x_0|=R}\|u\|_{L^\infty(x_0-1,x_0+1)} \ge \exp(-CR\log R) \] for all sufficiently large $R$. After the Caffarelli--Silvestre extension and the substitution $y=z^2/2$, the equation becomes a Grushin equation with a weak Robin condition on the degeneracy line. The corresponding angular operator has the arithmetic spectrum $κ_n=2n+\tfrac12$ after half-density conjugation. The central spectral estimate is \[ \sup_{ξ\in\mathbb R} \bigl\|C\bigl((τ+iξ)^2-L_0\bigr)^{-1}C^*\bigr\| \le Cτ^{-1/2} \] for parameters separated from the angular lattice. It yields a linear-weight Carleman estimate for measurable Robin feedback with absorption threshold $τ\ge C(1+\|V\|_\infty^2)$. Quantitative inward propagation, fixed-scale Grushin-ball propagation, and an interior-cylinder interpolation estimate then transfer bulk non-vanishing to the boundary. The Landis rescaling converts the local potential dependence $C\|q\|_\infty^2$ into the global rate $CR\log R$.
Comments38 pages
