Dispersive estimates for higher dimensional Schrödinger operators with threshold eigenvalues. II: The even dimensional case

Abstract

We investigate $L^1({\mathbb{R}}^n)\to L^{\infty }({\mathbb{R}}^n)$ dispersive estimates for the Schrödinger operator $H=-\Delta +V$ when there is an eigenvalue at zero energy in even dimensions $n\ge 6$. In particular, we show that if there is an eigenvalue at zero energy then there is a time dependent, rank one operator $F_t$ satisfying $\Vert F_t{\Vert }_{L^1\to L^{\infty }}\lesssim |t{|}^{2-\frac{n}{2}}$ for $|t|>1$ such that

With stronger decay conditions on the potential it is possible to generate an operator-valued expansion for the evolution, taking the form

with $A_{-2}$ and $A_{-1}$ mapping $L^1({\mathbb{R}}^n)$ to $L^{\infty }({\mathbb{R}}^n)$ while $A_0$ maps weighted $L^1$ spaces to weighted $L^{\infty }$ spaces. The leading-order terms $A_{-2}$ and $A_{-1}$ are both finite rank, and vanish when certain orthogonality conditions between the potential $V$ and the zero energy eigenfunctions are satisfied. We show that under the same orthogonality conditions, the remaining $|t{|}^{-\frac{n}{2}}{A}_0$ term also exists as a map from $L^1({\mathbb{R}}^n)$ to $L^{\infty }({\mathbb{R}}^n)$, hence $e^{itH}{P}_{ac}(H)$ satisfies the same dispersive bounds as the free evolution despite the eigenvalue at zero.