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

## TL;DR

This paper investigates the fourth order Schr"odinger operator in three dimensions, classifies zero energy resonances, and analyzes their impact on dispersive decay rates, revealing differences from the classical Schr"odinger case.

## Contribution

It provides a complete classification of zero energy resonances for the fourth order Schr"odinger operator and examines their effects on dispersive estimates, including necessary modifications for certain resonances.

## Key findings

- Attains the $|t|^{-3/4}$ decay rate in dispersive bounds.
- Identifies and constructs finite rank corrections for some resonances.
- Shows that resonance classification and effects differ from the standard Schr"odinger operator.

## Abstract

We study the fourth order Schr\"odinger operator $H=(-\Delta)^2+V$ 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 $L^1\to L^\infty$ dispersive bounds. In all cases, we show that the natural $|t|^{-\frac34}$ 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 $-\Delta+V$.