Time integrable weighted dispersive estimates for the fourth order Schr\"odinger equation in three dimensions

TL;DR

This paper establishes enhanced time decay estimates for the solution operator of the fourth order Schrödinger equation in three dimensions, especially under conditions of no eigenvalues or resonances, and analyzes the case with a mild resonance at zero energy.

Contribution

It provides new decay bounds for the fourth order Schrödinger operator, improving upon free case results, and derives operator expansions in the presence of a zero-energy resonance.

Findings

Solution operator decays as |t|^{-5/4} in weighted spaces.

Improved decay rates over the free case in two directions.

Operator expansion for mild zero-energy resonance case.

Abstract

We consider the fourth order Schr\"odinger 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 $|t|^{-\frac54}$ 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 $e^{-itH}P_{ac}(H)=t^{-\frac34} A_0+t^{-\frac54}A_1$ where $A_0:L^1\to L^\infty$ is an operator of rank at most four and $A_1$ maps between polynomially weighted spaces.