Abstract. We study time decay estimates of the fourth-order Schrödinger operatorWe 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 −itH P ac under suitable spectrum assumptions of H. Based on Jensen-Kato type decay estimate and local decay estimate, we obtain the L 1 → L ∞ estimate of e −itH P ac in 3-dimension by Ginibre argument, and also establish the endpoint global Strichartz estimates of e −itH P ac 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.

This paper is mainly devoted to study time decay estimates of the higher-order Schrödinger type operator H = (−∆) m + V (x) in R n for n > 2m and m ∈ 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 Schrödinger 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 embedding eigenvalues is given for (−∆) m + V (x) with a repulsive potential.Date: September 12, 2019.

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We study the L 1 −L ∞ dispersive estimate of the inhomogeneous fourth-order Schrödinger operator H = ∆ 2 − ∆ + V(x) with zero energy obstructions in R 3 . For the related propagator e −itH , we prove that for 0 < t ≤ 1, then e −itH P ac (H) satisfies the |t| −3/4 -estimate. For t > 1, we prove that: 1) if zero is a regular point of H, then e −itH P ac (H) satisfies the |t| −3/2 -dispersive estimate. 2) if zero is a resonance of H, there exists a time dependent operator F t such that e −itH P ac (H) − F t satisfies the |t| −3/2 -dispersive estimate. 3) if zero is a resonance and / or an eigenvalue of H, then there exists a time dependent operator G t such that e −itH P ac (H) − G t satisfies the |t| −3/2 -dispersive estimate. Here F t and G t satisfy |t| −1/2 -dispersive estimates.

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