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 $n\\geq 6$. In particular, we show that if there is an eigenvalue at zero energy then there is a time dependent, rank one operator $Ft$ satisfying $|F_t|{L^1\\to L^\\infty} \\lesssim |t|^{2-\\frac {n}{2}}$ for $|t|>1$ such that $$ |e^{itH}P{ac}-F_t|{L^1\\to L^\\infty} \\lesssim |t|^{1-\\frac {n}{2}},\\,\\,\\,\\,\\,\\text{ for } |t|>1. $$ With stronger decay conditions on the potential it is possible to generate an operator-valued expansion for the evolution, taking the form $$ e^{itH} P{ac}(H)=|t|^{2-\\frac {n}{2}}A{-2}+ |t|^{1-\\frac {n}{2}} A\_{-1}+|t|^{-\\frac {n}{2}}A_0, $$ with $A{-2}$ and $A{-1}$ mapping $L^1(\\mathbb R^n)$ to $L^\\infty(\\mathbb R^n)$ while $A0$ 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.