A spectral mapping theorem and invariant manifolds for nonlinear Schrodinger equations

A spectral mapping theorem is proved that resolves a key problem in applyin g invariant manifold theorems to nonlinear Schrodinger type equations. The theorem is applied to the operator that arises as the linearization of the equation around a standing wave solution. We cast the problem in the contex t of space-dependent nonlinearities that arise in optical waveguide problem s. The result is, however, more generally applicable including to equations in higher dimensions and even systems. The consequence is that stable, uns table, and center manifolds exist in the neighborhood of a (stable or unsta ble) standing wave, such as a waveguide mode, under simple and commonly ver ifiable spectral conditions.