Dispersive Estimates for higher dimensional Schrödinger Operators with threshold eigenvalues I: The odd 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 and $n\ge 5$ is odd. 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 $$\Vert e^{itH}{P}_{ac}-F_t{\Vert }_{L^1\to L^{\infty }}\lesssim |t{|}^{1-\frac{n}{2}},\text{ for }|t|>1.$$ 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 $A_{-1}$ finite rank operators 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 $A_{-1}$ 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.

To appear in J. Funct. Anal

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