Weighted dispersive estimates for two-dimensional Schr\"odinger operators with Aharonov-Bohm magnetic field

Abstract

We consider two-dimensional Schr\"odinger operators $H$ with Aharonov-Bohm magnetic field and an additional electric potential. We obtain an explicit leading term of the asymptotic expansion of the unitary group $e^{-i t H}$ for $t\to\infty$ in weighted $L^2$ spaces. In particular, we show that the magnetic field improves the decay of $e^{-i t H}$ with respect to the unitary group generated by non-magnetic Schr\"odinger operators, and that the decay rate in time is determined by the magnetic flux.