On the Fourth order Schr\"odinger equation in three dimensions: dispersive estimates and zero energy resonances

We study the fourth order Schr"odinger operator $H=(-\mathrm{\Delta }{)}^2+V$ for a short range potential in three space dimensions. We provide a full classification of zero energy resonances and study the dynamic effect of each on the $L^1\to L^{\mathrm{\infty }}$ dispersive bounds. In all cases, we show that the natural $\mathrm{\mid }t{\mathrm{\mid }}^{-\frac{3}{4}}$ decay rate may be attained, though for some resonances this requires subtracting off a finite rank term, which we construct and analyze. The classification of these resonances, as well as their dynamical consequences differ from the Schr"odinger operator $-\mathrm{\Delta }+V$.