Time Decay Estimates for Fourth-Order Schrödinger Operators in Dimension Three

Abstract

This paper is concerned with the time decay estimates of the fourth order Schrödinger operator $H = ∆^2+V (x)$ 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 $e^{−it H}$ with a regular term $H^{α/4}:$

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 $H^{α/4}$ will be interesting in the well-posedness and scattering of nonlinear fourth order Schrödinger equations with potentials.