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

### Michael Goldberg

University of Cincinnati, USA### William R. Green

Rose-Hulman Institute of Technology, Terre Haute, USA

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## 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 $F_t$ satisfying $\|F_t\|_{L^1\to L^\infty} \lesssim |t|^{2-\frac {n}{2}}$ 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.

## Cite this article

Michael Goldberg, William R. Green, Dispersive estimates for higher dimensional Schrödinger operators with threshold eigenvalues. II: The even dimensional case. J. Spectr. Theory 7 (2017), no. 1, pp. 33–86

DOI 10.4171/JST/155