Abstract

We consider the Schrödinger operators HV=−Δg+V with singular potentials V on general n-dimensional Riemannian manifolds and study the eigenvalues and eigenfunctions under this perturbation. These singular potentials appear naturally in physics, most notably the Coulomb potential |x|−1. Sogge and the first author [14] proved the sharp Weyl laws for these HV with potentials in the Kato class, which is the minimal assumption to ensure that HV is essentially self-adjoint and bounded from below and the eigenfunctions of HV are bounded. Later, Frank-Sabin [9] studied the problem on the pointwise Weyl laws for these HV in three dimensions by extending the method of Avakumović [2], while it is unknown how to reconstruct this argument in other dimensions. In this paper, we completely solve this problem in any dimensions by using a different argument. First, we establish the pointwise Weyl law for potentials in the Kato class on any n-dimensional manifolds. This extends the 3-dimensional results of Frank-Sabin [9] by a different method. Second, we prove that the pointwise Weyl law with the standard sharp error term O(λn−1) holds for potentials in Ln(M). This extends the classical results for smooth potentials by Avakumović [2], Levitan [18] and Hörmander [12] to critically singular potentials. In three dimensions, this L3 condition also naturally appears in Boccato-Brennecke-Cenatiempo-Schlein [6] on the ground state energy of the Hamilton operator in the Gross-Pitaevskii regime. These two results are sharp, and our proof exploits Li-Yau's heat kernel bounds and Blair-Sire-Sogge's eigenfunction estimates.

Full Text
Paper version not known

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.