<p>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 <i>intertwining operators</i>, 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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>. 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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-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 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-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.</p>

Intertwining Operators Beyond the Stark Effect

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2026年4月4日

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Luca Fanelli,
Xiaoyan Su,
Ying Wang,
Junyong Zhang,
Jiqiang Zheng

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 (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.