Mathematical Research Letters

Volume 29 (2022)

Number 3

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

Pages: 727 – 762

DOI: https://dx.doi.org/10.4310/MRL.2022.v29.n3.a5

Authors

Xiaoqi Huang (Department of Mathematics, Johns Hopkins University, Baltimore, Maryland, U.S.A.)

Christopher D. Sogge (Department of Mathematics, Johns Hopkins University, Baltimore, Maryland, U.S.A.)

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 $1/p-\textrm{loss} \: L^p_t \, L^q_x \, (I \times M)$-Strichartz estimates hold for $e^{-itH_V}$ when $H_V = -\Delta_g + V(x)$ with $V \in L^{n/2} (M)$ if $n \geq 3$ or $V \in L^{1+\delta} (M)$, $\delta \gt 0$, 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 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.

Full Text (PDF format)

Received 16 November 2020

Accepted 1 June 2021

Published 30 November 2022