二次阿哈罗诺夫－玻姆非线性薛定谔方程的长程渐近行为
Long Range Asymptotics for the Quadratic Aharonov-Bohm NLS
AI总结：
该研究刻画了二次阿哈罗诺夫－玻姆非线性薛定谔方程解的长程渐近行为，明确了保持在哈密顿量定义域内的剖面条件及余项速率，结果具有尖锐性。
AI中文摘要：
我们研究二维空间上方程$i\partial_tu=H_\alpha u+\lambda|u|u$解的长程行为，其中$H_\alpha$是具有单个极点的阿哈罗诺夫－玻姆哈密顿量的Friedrichs实现。长程近似的对数相位可能会将某个剖面推出$H_\alpha$的定义域。我们通过阶数$\le \frac{1}{2}$的边界迹在0处的消失，刻画了保持在算子定义域内的剖面；在半通量$\alpha=1/2$时，没有非零迹留存。然而，$H_\alpha$全定义域内每个具有小$L^\infty$振幅的剖面，都确定了一个带有修正终态的唯一全局解，其余项速率对所有$0<b<1/2+\nu_\alpha$为$t^{-b}$，其中$\nu_\alpha=\min\{\alpha,1-\alpha\}$。对于满足消失迹条件的剖面，速率改进为对所有$0<b<1$。该结果是尖锐的，即若$\alpha\neq \frac{1}{2}$，我们可构造出误差大小为$t^{-1/2-\nu_\alpha}\log t$的剖面，排除了所有更快的速率。上界来自不属于$L^2$的残差的延迟Strichartz估计；下界是通过汉克尔变换的显式计算得到的。对于更光滑的剖面，我们还计算了一阶修正，其给出的余项速率满足$1<b<2$。
英文摘要：
We study the long range behavior of solutions to $i\partial_tu=H_αu+λ|u|u$ on $\mathbb R^2$, where $H_α$ is the Friedrichs realization of the Aharonov-Bohm Hamiltonian with a single pole. The logarithmic phase of the long range ansatz may push a profile out of the domain of $H_α$. We characterize profiles that stay in the operator domain by the vanishing of boundary traces at 0 of order $\le \frac 12$; at half flux $α=\frac 12$, no nonzero trace survives. However, every profile in the full domain of $H_α$ with small $L^\infty$ amplitude determines a unique global solution with a modified final state, with a remainder rate $t^{-b}$ for all $0<b<1/2+ν_α$, $ν_α=\min\{α,1-α\}$. For profiles satisfying the vanishing trace condition, the rate improves to every $0<b<1$. This result is sharp in the sense that, if $α\neq \frac 12$, we can construct profiles with an error of size $t^{-1/2-ν_α}\log t$, ruling out all faster rates. The upper bound comes from a retarded Strichartz estimate for a residual that is not in $L^2$; the lower bound is an explicit calculation via Hankel transforms. For smoother profiles we also compute the first correction, which gives remainder rates with $1<b<2$.
