Decay estimates for bi-Schr\"odinger operators in dimension one

TL;DR

This paper establishes sharp decay estimates for bi-Schr"odinger operators in one dimension, analyzing resolvent asymptotics at zero energy and deriving Strichartz estimates for related fourth-order Schr"odinger equations.

Contribution

It provides the first detailed analysis of zero resonance effects on decay rates for bi-Schr"odinger operators in one dimension, including resolvent expansions and decay estimates.

Findings

Zero resonances do not alter the optimal decay rate in 1D.

Resolved asymptotic expansions of the resolvent at zero energy.

Derived Strichartz estimates for fourth-order Schr"odinger equations.

Abstract

This paper is devoted to study the time decay estimates for bi-Schr\"odinger operators $H=\Delta^{2}+V(x)$ in dimension one with decaying potentials $V(x)$. We first deduce the asymptotic expansions of resolvent of $H$ at zero energy threshold without/with the presence of resonances, and then characterize these resonance spaces corresponding to different types of zero resonance in suitable weighted spaces $L_s^2({\mathbf{R}})$. Next we use them to establish the sharp $L^1-L^\infty$ decay estimates of Schr\"odinger groups $e^{-itH}$ generated by bi-Schr\"odinger operators also with zero resonances. As a consequence, Strichartz estimates are obtained for the solution of fourth-order Schr\"odinger equations with potentials for initial data in $L^2({\mathbf{R}})$. In particular, it should be emphasized that the presence of zero resonances does not change the optimal time decay rate of $e^{-itH}$ in dimension one, except at requiring faster decay rate of the potential.