Quasimode and Strichartz estimates for time-dependent Schrödinger equations with singular potentials

Abstract

We generalize the Strichartz estimates for Schrödinger operators on compact manifolds of Burq, Gérard and Tzvetkov [10] by allowing critically singular potentials $V$. Specifically, we show that their $1/p$--loss $L_t^p{L}_x^q(I\times M)$-Strichartz estimates hold for $e^{-it{H}_V}$ when $H_V=-{\Delta }_g+V(x)$ with $V\in L^{n/2}(M)$ if $n\ge 3$ or $V\in L^{1+\delta }(M)$, $\delta >0$, if $n=2$, with $(p,q)$ being as in the Keel-Tao theorem and $I\subset \mathbb{R}$ a bounded interval. We do this by formulating and proving new "quasimode" estimates for scaled dyadic unperturbed Schrödinger operators and taking advantage of the the fact that $1/q^{\prime }-1/q=2/n$ for the endpoint Strichartz estimates when $(p,q)=(2,2n/(n-2))$. We also show that the universal quasimode estimates that we obtain are saturated on {\em any} compact manifolds; however, we suggest that they may lend themselves to improved Strichartz estimates in certain geometries using recently developed "Kakeya-Nikodym" techniques developed to obtain improved eigenfunction estimates assuming, say, negative curvatures.

Revised version to appear in Math. Research Letters