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We investigate dispersive estimates for the Schrödinger operator when there is an eigenvalue at zero energy in even dimensions . In particular, we show that if there is an eigenvalue at zero energy then there is a time dependent, rank one operator satisfying for 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 and mapping to while maps weighted spaces to weighted spaces. The leading-order terms and are both finite rank, and vanish when certain orthogonality conditions between the potential and the zero energy eigenfunctions are satisfied. We show that under the same orthogonality conditions, the remaining term also exists as a map from to , hence 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. 33–86DOI 10.4171/JST/155