Improved time-decay for a class of many-magnetic Schrödinger flows

Abstract

Consider the doubled magnetic Schrödinger operatorHα,B0=(i∇−(B0|x|2+α|x|)(−x2|x|,x1|x|))2,x=(x1,x2)∈R2∖{0}, where B0|x|2(−x2|x|,x1|x|) stands for the homogeneous magnetic potential with B0>0 and α|x|(−x2|x|,x1|x|) is the well-known Aharonov-Bohm potential with α∈R∖Z. In this note, we obtain an improved time-decay estimate for the Schrödinger flow e−itHα,B0. The key ingredient is the dispersive estimate for e−itHα,B0, which was established in [28] recently. This work is motivated by L. Fanelli, G. Grillo and H. Kovařík [16] dealing with the scaling-critical electromagnetic potentials in two and higher dimensions.