Decay estimates for bi-Schrödinger operators in dimension one

2021/06/30 by Soffer, Avy, Wu, Zhao, Yao, Xiaohua
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2106.15966

Abstract

This paper is devoted to study the time decay estimates for bi-Schrödinger operators H=Δ²+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_s²(R). Next we use them to establish the sharp L¹-L^∞ decay estimates of Schrödinger groups e^-itH generated by bi-Schrödinger operators also with zero resonances. As a consequence, Strichartz estimates are obtained for the solution of fourth-order Schrödinger equations with potentials for initial data in L²(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.

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