Fourier transform of surface--carried measures of two-dimensional generic surfaces and applications

TL;DR

This paper proves sharp decay estimates for Fourier transforms of surface measures on generic 2D surfaces and applies these results to derive new spectral and scattering properties of elliptic and Schrödinger operators.

Contribution

It provides a simplified proof of decay estimates and extends their application to spectral theory and scattering for discrete Schrödinger operators.

Findings

Sharp decay estimates for Fourier transforms of surface measures

Strichartz and resolvent estimates for elliptic operators

New spectral and scattering results for lattice Schrödinger operators

Abstract

We give a simple proof of the sharp decay of the Fourier-transform of surface-carried measures of two-dimensional generic surfaces. The estimates are applied to prove Strichartz and resolvent estimates for elliptic operators whose characteristic surfaces satisfy the generic assumptions. We also obtain new results on the spectral and scattering theory of discrete Schr\"odinger operators on the cubic lattice.