Sharp Strichartz estimates for some variable coefficient Schr\"{o}dinger operators on $\mathbb{R}\times\mathbb{T}^2$

TL;DR

This paper establishes sharp Strichartz estimates for certain variable coefficient Schrödinger operators on a mixed real and torus domain, extending known results to more complex operators with variable coefficients.

Contribution

It proves that Strichartz estimates at the same regularity as constant coefficient cases hold for Schrödinger operators with specific variable coefficients in two dimensions.

Findings

Strichartz estimates are valid for variable coefficient Schrödinger operators.

Results extend to higher dimensions with appropriate adjustments.

Analysis covers operators with space-dependent Laplacians.

Abstract

In the first part of the paper we continue the study of solutions to Schr\"odinger equations with a time singularity in the dispersive relation and in the periodic setting. In the second we show that if the Schr\"odinger operator involves a Laplace operator with variable coefficients with a particular dependence on the space variables, then one can prove Strichartz estimates at the same regularity as that needed for constant coefficients. Our work presents a two dimensional analysis, but we expect that with the obvious adjustments similar results are available in higher dimensions.