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Decay estimates and Strichartz estimates of fourth-order Schrödinger operator. (English) Zbl 1379.58013

Summary: We study time decay estimates of the fourth-order Schrödinger operator \(H=(-\Delta)^2+V(x)\) in \(\mathbb{R}^d\) for \(d=3\) and \(d \geq 5\). 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 \(e^{-itH}Pac\) under suitable spectrum assumptions of \(H\). Based on Jensen-Kato type decay estimate and local decay estimate, we obtain the \(L^1 \to L^\infty\) estimate of \(e^{-itH}Pac\) in 3-dimension by Ginibre argument, and also establish the endpoint global Strichartz estimates of \(e^{-itH}Pac\) for \(d \geq 5\). 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 \(H\).

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
42B15 Multipliers for harmonic analysis in several variables
35P15 Estimates of eigenvalues in context of PDEs
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
47F05 General theory of partial differential operators
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