Dynamical restriction for Schrödinger equations

We prove a dynamical restriction principle, asserting that every restriction estimate satisfied by the Fourier transform in ${\mathbb{R}}^d$ 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 $L^p({\mathbb{R}}^d)$, for $p$ in a suitable range of exponents, the solution $u(t,\cdot )$ (for each fixed $t$, with the exception of certain particular values) can be meaningfully restricted to compact curved submanifolds of ${\mathbb{R}}^d$. The underlying property responsible for this phenomenon is the boundedness of the propagator $L^p\to (\mathcal{F}{L}^p{)}_{\mathrm{l}\mathrm{o}\mathrm{c}}$, with $1\le p\le 2$, which is derived from almost diagonalization and dispersive estimates in function spaces defined in terms of wave packet decompositions in phase space.