Dynamical restriction for Schrödinger equations

2025/05/18 by Nicola, Fabio
#Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2505.12527

Abstract

We prove a dynamical restriction principle, asserting that every restriction estimate satisfied by the Fourier transform in 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(ℝ^d), 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 ℝ^d. The underlying property responsible for this phenomenon is the boundedness of the propagator L^p→(FL^p)_{\rm 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.

Citations

• A dynamical Amrein-Berthier uncertainty principle
• Fourier restriction for Schatten class operators and functions on phase space
• Fractional Fourier transforms, harmonic oscillator propagators and Strichartz estimates on Pilipovic and modulation spaces
• An introduction to the Gabor wave front set
• The Gabor wave front set of compactly supported distributions
• Microlocal properties of Shubin pseudodifferential and localization\n operators
• The equality of the homogeneous and the Gabor wave front set
• The Gabor wave front set
• Some recent progress on the Restriction conjecture

Related

• Strichartz estimates and Fourier restriction theorems on the Heisenberg\n group
• Fourier Restriction: From Linear Restriction to Multilinear Restriction
• Long time dynamics of Schrödinger and wave equations on flat tori
• Restriction of Schrödinger eigenfunctions to submanifolds
• Discrete Fourier restriction associated with Schrodinger equations