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

TL;DR

This paper derives explicit asymptotic decay estimates for the Schr"odinger evolution operator in two dimensions with Aharonov-Bohm magnetic field, showing magnetic flux influences decay rates in weighted spaces.

Contribution

It provides the first explicit leading term for the asymptotic expansion of the Schr"odinger propagator with Aharonov-Bohm magnetic field in weighted $L^2$ spaces.

Findings

Magnetic field enhances decay of the Schr"odinger evolution.

Decay rate depends on magnetic flux.

Explicit asymptotic expansion obtained.

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.