Intertwining Operators Beyond the Stark Effect

Abstract 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 (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 $$L^p$$ L p . 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 $$L^p$$ L p -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 $$L^p$$ L p -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.