We study the long range behavior of solutions to i∂tu=Hαu+λ∣u∣u on R2
Nearby in the stack
, where
Hα
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
Hα
. We characterize profiles that stay in the operator domain by the vanishing of boundary traces at 0 of order
≤21
; at half flux
α=1/2
, no nonzero trace survives. However, every profile in the full domain of
Hα
with small
L∞
amplitude determines a unique global solution with a modified final state, with a remainder rate
t−b
for all
0<b<1/2+να
,
να=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=21
, we can construct profiles with an error of size
t−1/2−ναlogt
, ruling out all faster rates. The upper bound comes from a retarded Strichartz estimate for a residual that is not in
L2
; the lower bound is an explicit calculation via Hankel transforms. For smoother profiles we also compute the first correction, which gives remainder rates with