Smoothing effect and quantum-classical correspondence for the Schrödinger equation with confining potential

2024/12/02 by Antoine Prouff, Prouff, Antoine
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #Quantum Mechanics and Non-Hermitian Physics #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2412.01209

openalex publication_date 2024/12/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The smoothing effect states that solutions to the Schrödinger equation in the Euclidean space have, for almost-every time, a local-in-space improved regularity (gain of half a derivative in Sobolev spaces). In this note, we show that, for the Schrödinger equation with a sub-quadratic confining potential, the smoothing effect is equivalent to an escape rate estimate on the associated classical flow. The proof relies on an Egorov theorem proved in~\citeP:24Egorovinprep.

Related

• Smoothing effect for Schrödinger boundary value problems
• The Kato smoothing effect for Schrödinger equations with unbounded potentials in exterior domains
• Global-in-time smoothing effects for Schrödinger equations with inverse-square potentials
• Smoothing effect for the regularized Schrödinger equation with non controlled orbits
• On the Local Smoothing for the Schroedinger Equation