Long Range Asymptotics for the Quadratic Aharonov-Bohm NLS

Abstract

We study the long range behavior of solutions to $i{\partial }_{t}u=H_{\alpha }u+\lambda |u|u$ on ${\mathbb{R}}^2$, where $H_{\alpha }$ 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 $H_{\alpha }$. We characterize profiles that stay in the operator domain by the vanishing of boundary traces at 0 of order $\le \frac{1}{2}$; at half flux $\alpha =\frac{1}{2}$, no nonzero trace survives. However, every profile in the full domain of $H_{\alpha }$ with small $L^{\infty }$ amplitude determines a unique global solution with a modified final state, with a remainder rate $t^{-b}$ for all $0<b<1/2+{\nu }_{\alpha }$, ${\nu }_{\alpha }=\min \{\alpha ,1-\alpha \}$. For profiles satisfying the vanishing trace condition, the rate improves to every $0<b<1$. This result is sharp in the sense that, if $\alpha \ne \frac{1}{2}$, we can construct profiles with an error of size $t^{-1/2-{\nu }_{\alpha }}\log t$, 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 $1<b<2$.