A Spectral Multiplier Theorem associated with a Schr\"odinger Operator

TL;DR

This paper proves a spectral multiplier theorem for Schr"odinger operators in three dimensions using a novel approach with the Born series, enabling analysis without Gaussian heat kernel estimates, and applies it to nonlinear PDEs.

Contribution

Introduces a new spectral multiplier theorem for Schr"odinger operators using the Born series, expanding analysis capabilities beyond Gaussian heat kernel assumptions.

Findings

Established a spectral multiplier theorem for Schr"odinger operators in D.

Provided an explicit integral representation for spectral multipliers.

Proved local well-posedness for a 3D quintic nonlinear Schrodinger equation with potential.

Abstract

We establish a spectral multiplier theorem associated with a Schr\"odinger operator H=-\Delta+V(x) in \mathbb{R}^3. We present a new approach employing the Born series expansion for the resolvent. This approach provides an explicit integral representation for the difference between a spectral multiplier and a Fourier multiplier, and it allows us to treat a large class of Schr\"odinger operators without Gaussian heat kernel estimates. As an application to nonlinear PDEs, we show the local-in-time well-posedness of a 3d quintic nonlinear Schr\"odinger equation with a potential.