• Abstract

This paper is concerned with the time decay estimates of  the fourth order Schrodinger  operator $H=\Delta^{2}+V(x)$ in dimension three, where $V(x)$ is  a real valued decaying potential.   Assume that zero is  a regular point or the first kind resonance of $H$,  and $H$ has no positive eigenvalues, we established the following  time optimal decay estimates of  $e^{-itH}$ with a regular term $H^{\alpha/4}$:

 $$\|H^{\alpha/4}e^{-itH}P_{ac}(H)\|_{L^1-L^\infty}\lesssim |t|^{-\frac{3+\alpha}{4}}, \quad   0 \leq \alpha \leq 3.$$

When zero is the second or  third kind  resonance of $H$,   their decay will be significantly changed. We remark that such improved  time decay estimates with the extra regular term $H^{\alpha/4}$  will be interesting in the well-posedness and scattering of nonlinear fourth order Schrodinger equations with potentials.

• General Summary