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 $\|F_t\|_{L^1\to L^\infty } \lesssim 1$ such that \[ \|e^{itH}P_{ac}-F_t\|_{L^1\to L^\infty } \lesssim |t|^{-1}, \text { for } |t|>1.\] 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.