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