Dispersive estimates for inhomogeneous fourth-order Schrödinger operator in 3D with zero energy obstructions

Abstract

We study the L1−L∞ dispersive estimate of the inhomogeneous fourth-order Schrödinger operator H=Δ2−Δ+V(x) with zero energy obstructions in R3. For the related propagator e−itH, we prove that for 0<|t|≤1, then e−itHPac(H) satisfies the |t|−3∕4-dispersive estimate. For |t|>1, we prove that:(1) if zero is a regular point of H, then e−itHPac(H) satisfies the |t|−3∕2-dispersive estimate.(2) If zero is purely a resonance of H, there exists a time dependent operator Ft such that e−itHPac(H)−Ft satisfies the |t|−3∕2-dispersive estimate.(3) If zero is purely an eigenvalue or zero is both an eigenvalue and a resonance of H, then there exists a time dependent operator Gt such that e−itHPac(H)−Gt satisfies the |t|−3∕2-dispersive estimate. Here Ft and Gt satisfy the |t|−1∕2-dispersive estimate.