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- W3102725969 abstract "We generalize the Strichartz estimates for Schrodinger operators on compact manifolds of Burq, G'erard and Tzvetkov [10] by allowing critically singular potentials $V$. Specifically, we show that their $1/p$--loss $L^p_tL^q_x(Itimes M)$-Strichartz estimates hold for $e^{-itH_V}$ when $H_V=-Delta_g+V(x)$ with $Vin L^{n/2}(M)$ if $nge3$ or $Vin L^{1+delta}(M)$, $delta>0$, if $n=2$, with $(p,q)$ being as in the Keel-Tao theorem and $Isubset {mathbb R}$ a bounded interval. We do this by formulating and proving new quasimode estimates for scaled dyadic unperturbed Schrodinger operators and taking advantage of the the fact that $1/q'-1/q=2/n$ 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." @default.
- W3102725969 created "2020-11-23" @default.
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- W3102725969 date "2020-11-08" @default.
- W3102725969 modified "2023-09-26" @default.
- W3102725969 title "Quasimode and Strichartz estimates for time-dependent Schrodinger equations with singular potentials" @default.
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- W3102725969 doi "https://doi.org/10.48550/arxiv.2011.04007" @default.
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