标题： 二维情况下零能级存在障碍的薛定谔算子的色散估计 显示英文标题

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

摘要： 我们研究当零能量处存在障碍、共振或本征值时，薛定谔算子$H=-\Delta+V$的$L^1(\R^2)\to L^\infty(\R^2)$散射估计。特别是，我们证明零能量处存在 s 波共振不会破坏$t^{-1}$衰减率。我们还证明，如果零能量处存在 p 波共振或本征值，则存在一个时间依赖的算子$F_t$满足$\|F_t\|_{L^1\to L^\infty} \lesssim 1$，使得$$\|e^{itH}P_{ac}-F_t\|_{L^1\to L^\infty} \lesssim |t|^{-1}, \text{for} |t|>1.$$。我们还在零能量处存在本征值但没有共振的情况下，建立了具有$t^{-1}$衰减率的加权散射估计。 显示英文摘要

摘要： We investigate $L^1(\R^2)\to L^\infty(\R^2)$ dispersive estimates for the Schr\"odinger 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.