Strichartz Estimates for Schrödinger Equations with Variable Coefficients

TL;DR

This paper proves local-in-time Strichartz estimates for Schrödinger equations with asymptotically flat, non-trapping variable-coefficient perturbations of the Laplacian. The authors reduce the problem to a small perturbation of the flat Laplacian and construct a microlocal parametrix via the FBI transform, solving a complex-phase eikonal equation with careful outgoing/incoming decomposition. The core novelty lies in a detailed flow analysis, a Melin–Sjöstrand/Hörmander preparation framework, and precise phase-data estimates that together yield dispersion bounds and, via standard machinery, the full range of scaling-admissible Strichartz estimates. The work integrates microlocal analysis, complex-phase Fourier integral operator techniques, and delicate handling of nontrapping perturbations to extend Strichartz theory beyond compact perturbations and into slowly decaying variable-coefficient settings.

Abstract

We prove the (local in time) Strichartz estimates (for the full range of parameters given by the scaling unless the end point) for asymptotically flat and non trapping perturbations of the flat Laplacian in $\R^n$, $n\geq 2$. The main point of the proof, namely the dispersion estimate, is obtained in constructing a parametrix. The main tool for this construction is the use of the FBI transform.

Theorems & Definitions (118)

• Theorem I.0.1
• Theorem II.1.1
• Theorem II.1.2
• Lemma II.1.3
• Lemma II.1.4
• Theorem II.1.5
• Theorem II.2.1
• Lemma III.1.1
• Lemma III.1.2
• Proposition III.2.1