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

TL;DR

This paper investigates the dispersive properties of the fourth order Schrödinger operator in four dimensions, analyzing how zero energy resonances influence decay rates and providing a detailed classification of these resonances.

Contribution

It offers a comprehensive analysis of zero energy resonances for the fourth order Schrödinger operator and their impact on dispersive decay estimates in four dimensions.

Findings

$t^{-1}$ decay rate under regular zero energy conditions

Faster $t^{-1}( ext{log } t)^{-2}$ decay with logarithmic weights when thresholds are regular

Classification of zero energy resonances and their effects on decay

Abstract

We study the fourth order Schr\"odinger operator $H=(-\Delta)^2+V$ for a decaying potential $V$ in four dimensions. In particular, we show that the $t^{-1}$ decay rate holds in the $L^1\to L^\infty$ setting if zero energy is regular. Furthermore, if the threshold energies are regular then a faster decay rate of $t^{-1}(\log t)^{-2}$ is attained for large $t$, at the cost of logarithmic spatial weights. Zero is not regular for the free equation, hence the free evolution does not satisfy this bound due to the presence of a resonance at the zero energy. We provide a full classification of the different types of zero energy resonances and study the effect of each type on the time decay in the dispersive bounds.