Decay estimates for higher-order elliptic operators

Abstract

This paper is mainly devoted to the study of time decay estimates of the higher-order Schrödinger-type operator H = ( − Δ ) m + V ( x ) H=(-\Delta )^{m}+V(x) in R n \mathbf {R}^{n} for n > 2 m n>2m and m ∈ N m\in \mathbf {N} . For certain decay potentials V ( x ) V(x) , we first derive the asymptotic expansions of resolvent R V ( z ) R_{V}(z) near zero threshold with the presence of zero resonance or zero eigenvalue, as well as 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 Schrödinger propagator e − i t H e^{-itH} in the presence of zero resonance or zero eigenvalue. As a consequence, the endpoint Strichartz estimate and L p L^{p} -decay estimates can also be obtained. Finally, by a virial argument, a criterion on the absence of positive embedding eigenvalues is given for ( − Δ ) m + V ( x ) (-\Delta )^{m}+V(x) with a repulsive potential.