Dispersive estimates for Schrödinger operators in dimension two with obstructions at zero energy

Abstract

We investigate $L^1({\mathbb{R}}^2)\to L^{\infty }({\mathbb{R}}^2)$ dispersive estimates for the Schrödinger operator $H=-\Delta +V$ when there are obstructions, resonances or an eigenvalue, at zero energy. In particular, we show that the existence of an s-wave resonance at zero energy does not destroy the $t^{-1}$ decay rate. We also show that if there is a p-wave resonance or an eigenvalue at zero energy then there is a time dependent operator $F_t$ satisfying $\Vert F_t{\Vert }_{L^1\to L^{\infty }}\lesssim 1$ such that $$\Vert e^{itH}{P}_{ac}-F_t{\Vert }_{L^1\to L^{\infty }}\lesssim |t{|}^{-1},\text{for}|t|>1.$$ We also establish a weighted dispersive estimate with $t^{-1}$ decay rate in the case when there is an eigenvalue at zero energy but no resonances.

41 pages

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