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Intertwining Operators Beyond the Stark Effect
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DOI：10.1007/s00220-026-05600-w.png)
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Abstract
En 中文
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 Schr & ouml;dinger groups and can be responsible for the lack of dispersion (Fanelli et al. in Commun Math Phys 324:1033-1067, 2013; in Commun Math Phys 337:1515-1533, 2015; in J Spectr Theory 8:509-521, 2018). Recently, Miao et al. introduced in Miao et al. http://arxiv.org/abs/2405.02531 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>p$$\end{document}. 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, which is highly non trivial. In addition, we investigate the mapping properties between Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>p$$\end{document}-spaces of these operators. In 2D, we prove a complete result, for the Schr & ouml;dinger Hamiltonian with a (fixed) magnetic potential an electric potential, both scaling critical, allowing us to prove dispersive estimates, uniform resolvent estimates, and Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>p$$\end{document}-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.
Keywords:
L-P-BOUNDEDNESS
SCHRODINGER-OPERATORS
TIME-DECAY
WAVE-OPERATORS
OSCILLATORY INTEGRALS
POTENTIALS
EQUATIONS
Journal
C
IF:
2.6
Papers:
195
Citations:
0
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