Decay estimates for the Schrödinger operators with electro-magnetic potentials in dimension two with obstructions at zero energy

Abstract

In this paper, we present the first proof of the decay estimates for the Schrödinger operators with electro-magnetic potentials when there are obstructions, eigenvalue or resonances at zero energy in $\mathbb{R}^{2}$. Specifically, we establish the dispersive estimates $L^{1}\rightarrow L^{\infty}$ if there is regular point or first kind of resonance point at zero energy. For the other cases, i.e. zero is the second or third kind of resonance, we show the decay estimates with not $t^{-1}$ decay rate. To achieve this, we derive expansions for the resolvent with magnetic potentials respectively when $0$ belongs to four distinct types: regular point and three kinds of resonance points. We then estimate the resolvent and the oscillatory integral to establish the decay result.