Dynamical restriction for Schrödinger equations

We prove a dynamical restriction principle, asserting that every restriction estimate satisfied by the Fourier transform in Rd is also valid for the propagator of certain Schrödinger equations. We consider smooth Hamiltonians with an at most quadratic growth, and also a class of nonsmooth Hamiltonians, encompassing potentials that are Fourier transforms of complex (finite) Borel measures. Roughly speaking, if the initial datum belongs to Lp(Rd), for p in a suitable range of exponents, the solution u(t,⋅) (for each fixed t, with the exception of certain particular values) can be meaningfully restricted to compact curved submanifolds of Rd. The underlying property responsible for this phenomenon is the boundedness of the propagator Lp→(FLp)loc, with 1≤p≤2, which is derived from almost diagonalization and dispersive estimates in function spaces defined in terms of wave packet decompositions in phase space.