Decay estimates and Strichartz estimates of fourth-order Schrödinger operator

Abstract

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.

On the fourth order Schrödinger equation in four dimensions: Dispersive estimates and zero energy resonances

“…Similar dynamics are expected also for the fourth order Schrödinger equation, in the sense that zero energy obstructions should make the time decay slower. In fact, it was shown in [11] that in this case if zero is regular then the natural time decay γ(t) = |t| − d 4 is valid in dimensions d > 4, and is |t| − 1 2 for large t in d = 3. In particular, we note that the case of d = 4, and the case when zero energy is not regular in all dimensions were open until now.…”

Decay estimates for higher-order elliptic operators

“…For the fourth order Schrödinger operator (−∆) 2 + V , i.e. the case m = 2, the first two authors and the last author in [FSY18] established the Kato-Jensen decay estimates with the time decay rate is (1+|t|) −n/4 for n ≥ 5 and (1+|t|) −5/4 for n = 3 in the regular case. Recently, the authors in [FWY18] further studied Kato-Jensen decay estimates in non-regular case for d ≥ 5.…”

“…Hence it would be a natural problem to establish the L 1 − L ∞ estimates for higher order Schrödinger operators with some potentials. For H = (−∆) 2 + V , in [FSY18], the first two authors and the last author applied the Kato-Jensen decay estimates and local decay estimate to obtain the In the sequel, as a consequence of the Kato-Jensen decay estimates, we can establish the following L 1 ∩ L 2 − L ∞ + L 2 -decay estimate (Ginibre argument) in the presence of zero resonance or eigenvalue.…”

Time Asymptotic expansions of solution for fourth-order Schrödinger equation with zero resonance or eigenvalue

“…For the fourth-order Schrödinger operator H = (−∆) 2 + V , Soffer, the first and third author in [9] first gave the asymptotic expansions of the resolvent of fourth-order Schrödinger operator (−∆) 2 + V around zero threshold assume that zero is a regular point of H for d ≥ 5 and d = 3. Their results show that, in the regular case, the time decay rate of Kato-Jensen decay estimate is (1+|t|) −d/4 with d ≥ 5.…”

“…In the case d = 4, Green and Toprak in [14] derived the asymptotic expansion of the resolvent R V (z) near zero with the presence of resonance or eigenvalue. Indeed, with the help of higher energy decay estimates established in [9], the time decay rate of Kato-Jensen decay estimate is (1 + |t|) −1 if zero is a regular point of H = (−∆) 2 + V . While, the time decay rate is (ln |t|) −1 as t → ∞ if zero is purely an eigenvalue of H.…”