Strichartz estimates for Schr\"odinger equations with variable coefficients and unbounded potentials II. Superquadratic potentials

TL;DR

This paper establishes local-in-time Strichartz estimates with derivative loss for Schrödinger equations featuring variable coefficients and unbounded, superquadratic potentials, extending previous results and employing advanced microlocal analysis techniques.

Contribution

It generalizes previous Strichartz estimates to cases with variable coefficients and superquadratic potentials, improving upon earlier work by Yajima-Zhang.

Findings

Proves Strichartz estimates with derivative loss for nontrapping geometries.

Handles polynomially growing superquadratic potentials.

Utilizes microlocal techniques and semiclassical parametrices.

Abstract

In this paper we prove local-in-time Strichartz estimates with loss of derivatives for Schr\"odinger equations with variable coefficients and potentials, under the conditions that the geodesic flow is nontrapping and potentials grow polynomially at infinity. This is a generalization to the case with variable coefficients and improvement of the result by Yajima-Zhang. The proof is based on microlocal techniques including the semiclassical parametrix for a time scale depending on a spatial localization and the Littlewood-Paley type decomposition with respect to both of space and frequency.