INTERTWINING OPERATORS BEYOND THE STARK EFFECT

作     者：Fanelli, Luca Su, Xiaoyan Wang, Ying Zhang, Junyong Zheng, Jiqiang 

作者机构：Ikerbasque Basque Foundation for Science Departamento de Matemáticas Universidad del País Vasco Euskal Herriko Unibertsitatea PV/EHU Leioa48940 Spain BCAM - Basque Center for Applied Mathematics Bilbao48009 Spain Loughborough University Loughborough United Kingdom Basque Center for Applied Mathematics Bilbao Spain Department of Mathematics Beijing Institute of Technology Beijing100081 China Institute of Applied Physics and Computational Mathematics National Key Laboratory of Computational Physics Beijing100088 China 

出 版 物：《arXiv》 (arXiv)

年 卷 期：2024年

核心收录：

主　　题：Eigenvalues and eigenfunctions 

摘      要：The main mathematical manifestation of the Stark effect in quantum mechanics is the shift and the formation of clusters of eigenvalues when a spherical Hamiltonian is perturbed by lower order terms. Understanding this mechanism turned out to be fundamental in the description of the large-time asymptotics of the associated Schrödinger groups and can be responsible for the lack of dispersion [14, 16, 17]. Recently, Miao, Su, and Zheng introduced in [31] a family of spectrally projected intertwining operators, reminiscent of the Kato’s wave operators, in the case of constant perturbations on the sphere (inverse-square potential), and also proved their boundedness in Lp. Our aim is to establish a general framework in which some suitable intertwining operators can be defined also for non constant spherical perturbations in space dimensions 2 and higher. In addition, we investigate the mapping properties between Lp-spaces of these operators. In 2D, we prove a complete result, for the Schrödinger Hamiltonian with a (fixed) magnetic potential an electric potential, both scaling critical, allowing us to prove dispersive estimates, uniform resolvent estimates, and Lp-bounds of Bochner–Riesz means. In higher dimensions, apart from recovering the example of inverse-square potential, we can conjecture a complete result in presence of some symmetries (zonal potentials), and open some interesting spectral problems concerning the asymptotics of eigenfunctions. © 2024, CC BY.