Maybe I don't understand why people keep bringing up bases; I would rather talk in terms of functions on configuration space. At least I'm sure I know what "local" means there.

The Schroedinger equation as a differential equation on functions on the momentum configuration space is exactly the same as on functions on position configuration space: you just replace q with p (and maybe signs). Switching p and q will make the Hamiltonian look different. But it's still a Hamiltonian in classical mechanics.