Time Asymptotic expansions of solution for fourth-order Schrödinger equation with zero resonance or eigenvalue

Abstract

In this paper, we first deduce the asymptotic expansions of resolvent of $H=(-\Delta {)}^2+V$ with the presence of resonance or eigenvalue at the degenerate zero threshold for $d\ge 5$. In particular, we identify these resonance spaces for full kinds of zero resonances. As a consequence, we then establish the {\it time asymptotic expansions} and {\it Kato-Jensen estimates} for the solution of fourth-order Schrödinger equation under the presence of zero resonance or eigenvalue.

35 pages. This is an updated version, where more backgrounds and comments are added