Decay estimates for higher order elliptic operators

TL;DR

This paper investigates the decay behavior of solutions to higher-order Schrödinger operators with potential in higher dimensions, deriving resolvent asymptotics, decay estimates, and criteria for embedded eigenvalues.

Contribution

It provides new asymptotic expansions of the resolvent near zero, analyzes the effects of zero resonance/eigenvalues on decay rates, and establishes decay and Strichartz estimates for higher-order Schrödinger operators.

Findings

Derived resolvent asymptotics near zero threshold.

Established decay estimates in the presence of zero resonance/eigenvalues.

Provided criteria for absence of positive embedded eigenvalues.

Abstract

This paper is mainly devoted to study time decay estimates of the higher-order Schr\"{o}dinger type operator $H=(-\Delta)^{m}+V(x)$ in $\mathbf{R}^{n}$ for $n>2m$ and $m\in\mathbf{N}$. For certain decay potentials $V(x)$, we first derive the asymptotic expansions of resolvent $R_{V}(z)$ near zero threshold with the presence of zero resonance or zero eigenvalue, as well identify the resonance space for each kind of zero resonance which displays different effects on time decay rate. Then we establish Kato-Jensen type estimates and local decay estimates for higher order Schr\"odinger propagator $e^{-itH}$ in the presence of zero resonance or zero eigenvalue. As a consequence, the endpoint Strichartz estimate and $L^{p}$-decay estimates can also be obtained. Finally, by a virial argument, a criterion on the absence of positive embedded eigenvalues is given for $(-\Delta)^{m}+V(x)$ with a repulsive potential.