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

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.