指数波积分算子在具有高度奇异势的非线性薛定谔方程中的最优误差界
Optimal error bounds on the exponential wave integrator for nonlinear Schrödinger equations with highly singular potential
AI总结：
本文研究了指数波积分算子在具有高度奇异势的非线性薛定谔方程中的误差估计，针对局部L^p空间中的奇异势，证明了在L^2正则性假设下，指数波积分算子的L^2范数收敛性最优，并首次在三维L^2势情况下达到最优收敛。
摘要：
我们建立了指数波积分算子（EWI）在具有高度奇异势的非线性薛定谔方程（NLSE）中的误差估计，该估计适用于R^d空间中1≤d≤3的情况。我们的结果处理了局部L^p空间中的奇异势，其中p> d/2且p≥1，这几乎是最弱的正则性要求，以确保NLSE的适定性。首先，对于L^p_loc势能，当p>2时，我们建立了指数波积分算子在L^2范数下的最优一阶收敛性，当p=2时，收敛阶稍降至1^-。据我们所知，三维L^2势能下的最优一阶收敛性在文献中首次被提出。此类误差界的优势是双重的：(i) 在给定的L^2正则性假设下，指数波积分算子（及其高阶版本）的一阶L^2范数收敛性是最佳的；(ii) 为了实现指数波积分算子的一阶L^2范数收敛性，这样的假设是最佳的。对于更奇异的局部L^p空间中的势能，其中d/2 < p < 2且p≥1，我们证明了L^2范数收敛性在d=1,2时为(1-α)-阶，在d=3时为(1-3/2 α)-阶，其中α: = d(1/p - 1/2)当d=1,2,3，p>1，且α: = 1/2^+当d=1，p=1。值得注意的是，这一结果将误差估计推到了与非线性薛定谔方程适定性阈值正则性相匹配的势能阈值正则性，这也是首次实现。证明中采用了两个主要成分：(i) 使用离散空间-时间Lebesgue空间和离散Strichartz估计来建立数值方案的稳定性；(ii) 使用正常形式变换和频率分解来获得最优误差界。
英文摘要：
We establish error estimates of the first-order exponential wave integrator (EWI) for the nonlinear Schrödinger equation (NLSE) with a highly singular potential in $\mathbb{R}^d$ with $1\leq d \leq 3$. Our results deal with singular potentials in $L^p_\text{loc}(\mathbb{R}^d)$ with $p>\frac{d}{2}$ and $p\geq 1$, which is (almost) the weakest regularity of the potential required by the well-posedness of the NLSE. First, for $L^p_\text{loc}$-potentials with $p>2$, we establish an optimal first-order $L^2$-norm convergence for the EWI, with the convergence order slightly reduced to $1^-$ when $p=2$. To the best of our knowledge, the optimal first-order convergence for the three-dimensional $L^2$-potential is for the first time in the literature. The optimality of such an error bound is two-fold: (i) the first-order $L^2$-norm convergence is optimal for the EWI (and its higher-order versions) under the given $L^2$-regularity assumption on the potential, and (ii) to achieve the first-order $L^2$-norm convergence for the EWI, such an assumption is optimally weak. For more singular potentials in $L^p_\text{loc}(\mathbb{R}^d)$ with $\frac{d}{2} < p < 2$ and $p\geq 1$, we prove that the $L^2$-norm convergence is (almost) of $(1-α)$-order when $d=1,2$, and of $(1-\frac{3}{2}α)$-order when $d=3$, where $α:=d(1/p - 1/2)$ when $d =1,2,3$, $p>1$ and $α:=\frac{1}{2}^+$ when $d=1$, $p=1$. Notably, this result pushes the error estimate to the threshold regularity of the potential that matches the threshold regularity for the well-posedness of the NLSE, which is also for the first time. Two main ingredients are adopted in the proof: (i) the use of discrete space-time Lebesgue spaces together with discrete Strichartz estimates to establish the stability of the numerical scheme, and (ii) the use of normal form transformation and frequency decompositions to obtain optimal error bounds.
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