Long Range Asymptotics for the Quadratic Aharonov-Bohm NLS

Abstract

We study the long range behavior of solutions to $i\partial_tu=H_\alpha u+\lambda|u|u$ 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 $\le \frac 12$; at half flux $\alpha=\frac 12$, 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 $\alpha\neq \frac 12$, 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 $1<b<2$.