A Spectral Multiplier Theorem associated with a Schrödinger Operator

Abstract

We establish a spectral multiplier theorem associated with a Schrödinger operator H=-Δ+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 Schrödinger 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 Schrödinger equation with a potential.

References in corpus (3)

• Dispersive estimates for Schrodinger operators in dimensions one and three
• Structure of wave operators in R^3
• Local-in-time well-posedness for Nonlinear Schroedinger equations with potentials