Dispersive estimates for inhomogeneous fourth-order Schr\"odinger operator in 3D with zero energy obstructions

TL;DR

This paper establishes dispersive estimates for the inhomogeneous fourth-order Schrödinger operator in 3D, analyzing how zero energy obstructions like resonances and eigenvalues affect the decay rates of the propagator over time.

Contribution

It provides new dispersive estimates for the operator considering zero energy obstructions, detailing different decay behaviors based on the spectral properties at zero energy.

Findings

For small times, the propagator decays as |t|^{-3/4}.

For large times, decay is |t|^{-3/2} under various spectral conditions.

Operators F_t and G_t capture additional decay properties related to zero energy obstructions.

Abstract

We study the $L^1-L^\infty$ dispersive estimate of the inhomogeneous fourth-order Schr\"{o}dinger operator $H=\Delta^{2}-\Delta+V(x)$ with zero energy obstructions in $\mathbf{R}^{3}$. For the related propagator $e^{-itH}$, we prove that for $0<t\leq 1$, then $e^{-itH}P_{ac}(H)$ satisfies the $|t|^{-3/4}$-estimate. For $t>1$, we prove that:\,\, 1) if zero is a regular point of $H$, then $e^{-itH}P_{ac}(H)$ satisfies the $|t|^{-3/2}$- dispersive estimate.\,\, 2) if zero is a resonance of $H$, there exists a time dependent operator $F_{t}$ such that $e^{-itH}P_{ac}(H)-F_{t}$ satisfies the $|t|^{-3/2}$- dispersive estimate.\,\, 3) if zero is a resonance and~/~or an eigenvalue of $H$, then there exists a time dependent operator $G_{t}$ such that $e^{-itH}P_{ac}(H)-G_{t}$ satisfies the $|t|^{-3/2}$- dispersive estimate. Here $F_{t}$ and $G_{t}$ satisfy $|t|^{-1/2}$-dispersive estimates.