On the Fourth order Schrödinger equation in three dimensions: dispersive estimates and zero energy resonances

Abstract

We study the fourth order Schrödinger operator $H=(-\Delta {)}^2+V$ for a short range potential in three space dimensions. We provide a full classification of zero energy resonances and study the dynamic effect of each on the $L^1\to L^{\infty }$ dispersive bounds. In all cases, we show that the natural $|t{|}^{-\frac{3}{4}}$ decay rate may be attained, though for some resonances this requires subtracting off a finite rank term, which we construct and analyze. The classification of these resonances, as well as their dynamical consequences differ from the Schrödinger operator $-\Delta +V$.

35 pages, submitted. arXiv admin note: text overlap with arXiv:1810.03678

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Cited by in corpus (7)

• On the $L^p$ boundedness of the Wave Operators for fourth order Schrödinger operators
• Kato smoothing, Strichartz and uniform Sobolev estimates for fractional operators with sharp Hardy potentials
• Time Asymptotic expansions of solution for fourth-order Schrödinger equation with zero resonance or eigenvalue
• Global Kato smoothing and Strichartz estimates for higher-order Schrödinger operators with rough decay potentials
• Dispersive estimates for inhomogeneous fourth-order Schrödinger operator in 3D with zero energy obstructions
• Decay estimates for higher order elliptic operators
• Decay estimates for fourth-order Schrödinger operators in dimension two