Journal of Spectral Theory


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Volume 7, Issue 1, 2017, pp. 33–86
DOI: 10.4171/JST/155

Published online: 2017-03-22

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

Michael Goldberg[1] and William R. Green[2]

(1) University of Cincinnati, USA
(2) Rose-Hulman Institute of Technology, Terre Haute, USA

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 $$\|e^{itH}P_{ac}-F_t\|_{L^1\to L^\infty} \lesssim |t|^{1-\frac {n}{2}},\,\,\,\,\,\text{ for } |t|>1.$$ 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 $A_0$ 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.

Keywords: Schrödinger operator, eigenvalue, dispersive estimate, resolvent

Goldberg Michael, Green William: Dispersive estimates for higher dimensional Schrödinger operators with threshold eigenvalues. II: The even dimensional case. J. Spectr. Theory 7 (2017), 33-86. doi: 10.4171/JST/155