Dynamical restriction for Schr\"odinger equations

TL;DR

This paper establishes that restriction estimates for Fourier transforms extend to solutions of certain Schrödinger equations with specific Hamiltonians, revealing new restriction phenomena for these quantum evolutions.

Contribution

It introduces a dynamical restriction principle for Schrödinger propagators with smooth and nonsmooth Hamiltonians, extending Fourier restriction estimates to these quantum dynamics.

Findings

Restriction estimates hold for Schrödinger solutions with quadratic growth Hamiltonians.

The propagator is bounded from L^p to localized Fourier-L^p spaces for 1≤p≤2.

Restriction to submanifolds is meaningful for solutions with initial data in certain L^p spaces.

Abstract

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 Schr\"odinger 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 $L^p\to(\mathcal{F}L^p)_{\rm loc}$, with $1\leq p\leq2$, which is derived from almost diagonalization and dispersive estimates in function spaces defined in terms of wave packet decompositions in phase space.