Weak dispersive estimates for fractional Aharonov-Bohm-Schr\"odinger groups

TL;DR

This paper establishes advanced dispersive and smoothing estimates for fractional Schrödinger equations influenced by Aharonov-Bohm magnetic fields in two dimensions, highlighting improvements when magnetic flux is non-integer.

Contribution

It introduces explicit flow representations and improves free estimates for fractional Schrödinger equations with Aharonov-Bohm fields, especially when flux is non-integer.

Findings

Proves local smoothing and energy decay estimates.

Derives weighted Strichartz inequalities.

Shows improved estimates for non-integer magnetic flux.

Abstract

We prove local smoothing, local energy decay and weighted Strichartz inequalities for fractional Schr\"odinger equations with a Aharonov-Bohm magnetic field in 2D. Explicit representations of the flows in terms of spherical expansions of the Hamiltonians are involved in the study. An improvement of the free estimate is proved, when the total flux of the magnetic field through the unit sphere is not an integer.