三维及以上空间中薛定谔方程Dirichlet解的消失阶
Vanishing orders of Dirichlet solutions to the Schrödinger equation in dimensions three and higher
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中文总结 AI 辅助
本文在三维及以上空间中构造薛定谔方程Dirichlet解，证明消失阶的尖锐上界为$C(1+\\|V\\|_\infty^{2/3})$，并指出Kukavica和Kenig的猜想界不可达到。
中文摘要 AI 辅助
设$B_3=B(0,3)\subset\mathbb{R}^3$。对于每个足够大的整数$k$，我们构造一个非零实函数$u_k\in C^2(\overline{B_3})$和一个实势$V_k\in L^\infty(B_3)$，使得$\Delta u_k=V_ku_k$，$u_k|_{\partial B_3}=0$，$\mathrm{ord}_0u_k=k$，且$\\|V_k\\|_\infty\le Ck^{3/2}$。因此，对于每个足够大的$N$，存在这样的Dirichlet解，其势范数小于$N$且消失阶至少为$cN^{2/3}$。该构造直接推广到更高维数和球面。结合先前结果，此例表明$C(1+\\|V\\|_\infty^{2/3})$是实值情形下消失阶的尖锐界。它也表明由Kukavica \cite{Kukavica1998}和Kenig \cite{Kenig2006}独立猜测的界在一般情况下不可达到。
英文摘要
Let $B_3=B(0,3)\subset\mathbb{R}^3$. For every sufficiently large integer $k$, we construct a nonzero real function $u_k\in C^2(\overline{B_3})$ and a real potential $V_k\in L^\infty(B_3)$ such that $Δu_k=V_ku_k$, $u_k|_{\partial B_3}=0$, $\mathrm{ord}_0u_k=k$, and $\|V_k\|_\infty\le Ck^{3/2}$. Consequently, for every sufficiently large $N$, there is such a Dirichlet solution with potential norm smaller than $N$ and vanishing order at least $cN^{2/3}$. The construction extends directly to higher dimensions and to spheres. Together with previous results, this example indicates that $C(1+\|V\|_\infty^{2/3})$ is the sharp bound for the vanishing order in the real-valued case. It also indicates that the bound conjectured independently by Kukavica \cite{Kukavica1998} and Kenig \cite{Kenig2006} is not attainable in general.
