Low frequency asymptotics and local energy decay for the Schrödinger equation

2021/10/15 by Julien Royer, Royer, Julien
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory in Mathematical Physics

paper · doi:10.48550/arxiv.2110.08081

openalex publication_date 2021/10/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove low frequency resolvent estimates and local energy decay for the Schrödinger equation in an asymptotically Euclidean setting. More precisely, we go beyond the optimal estimates by comparing the resolvent of the perturbed Schrödinger operator with the resolvent of the free Laplacian. This gives the leading term for the developpement of this resolvent when the spectral parameter is close to 0. For this, we show in particular how we can apply the usual commutators method for generalized resolvents and simultaneously for different operators. Finally, we deduce similar results for the large time asymptotics of the corresponding evolution problem.

Related

• Low frequency estimates and local energy decay for asymptotically euclidean Laplacians
• Semiclassical resolvent estimates for Holder potentials
• Low-energy resolvent estimates for slowly decaying attractive potentials
• Asymptotics in all regimes for the Schrödinger equation with time-independent coefficients
• Local energy decay for several evolution equations on asymptotically euclidean manifolds