Time integrable weighted dispersive estimates for the fourth order Schrödinger equation in three dimensions

Abstract

We consider the fourth order Schrödinger operator $H={\Delta }^2+V$ and show that if there are no eigenvalues or resonances in the absolutely continuous spectrum of $H$ that the solution operator $e^{-itH}$ satisfies a large time integrable $|t{|}^{-\frac{5}{4}}$ decay rate between weighted spaces. This bound improves what is possible for the free case in two directions; both better time decay and smaller spatial weights. In the case of a mild resonance at zero energy, we derive the operator-valued expansion $e^{-itH}{P}_{ac}(H)=t^{-\frac{3}{4}}{A}_0+t^{-\frac{5}{4}}{A}_1$ where $A_0:L^1\to L^{\infty }$ is an operator of rank at most four and $A_1$ maps between polynomially weighted spaces.

24 pages, submitted. Revised according to referee's comments. arXiv admin note: text overlap with arXiv:1905.02890

Cited by in corpus (1)

• On the $L^p$ boundedness of the Wave Operators for fourth order Schrödinger operators