Decay estimates and Strichartz estimates of fourth-order Schrödinger operator

Abstract

We study time decay estimates of the fourth-order Schrödinger operator H=(−Δ)2+V(x) in Rd for d=3 and d≥5. We analyze the low energy and high energy behaviour of resolvent R(H;z), and then derive the Jensen–Kato dispersion decay estimate and local decay estimate for e−itHPac under suitable spectrum assumptions of H. Based on Jensen–Kato type decay estimate and local decay estimate, we obtain the L1→L∞ estimate of e−itHPac in 3-dimension by Ginibre argument, and also establish the endpoint global Strichartz estimates of e−itHPac for d≥5. Furthermore, using the local decay estimate and the Georgescu–Larenas–Soffer conjugate operator method, we prove the Jensen–Kato type decay estimates for some functions of H.