Intertwining operators beyond the stark effect

by Dr Ying Wang (Basque Center for Applied Mathematics)

Friday Jun 19, 2026, 3:15 PM → 4:05 PM Europe/Berlin

Endenicher Allee 60, Seminarraum 0.011 (Mathezentrum)

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 [1, 2, 3]. Recently, Miao, Su, and Zheng introduced in [4] 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 $L^p$. 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, furthermore we investigate the mapping properties between $L^p$-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. 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. This is a jointed work with Luca Fanelli, Xiaoyan Su, Junyong Zhang and Jiqiang Zheng.

[1] T. Kato, Perturbation theory for linear operators, Springer-Verlag, Berlin, 1966.

[2] C. E. Kenig, A. Ruiz, and C. D. Sogge, Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators, Duke Math. J., 55 (1987), 329-347.

[3] B. G. Korenev, Bessel Functions and Their Applications, An International Series of Monographs in Mathematics, Taylor and Francis, 11 New Fetter Lane, London EC4P 4EE, 2002.

[4] G. N. Watson, A Treatise on the Theory of Bessel Functions. Second Edition, Cambridge University Press, 1944.