On the spectrum of nondegenerate magnetic Laplacians

Charles, Laurent

doi:10.2140/apde.2024.17.1907

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On the spectrum of nondegenerate magnetic Laplacians

Laurent Charles

Vol. 17 (2024), No. 6, 1907–1952

DOI: 10.2140/apde.2024.17.1907

Abstract

We consider a compact Riemannian manifold with a Hermitian line bundle whose curvature is nondegenerate. Under a general condition, the Laplacian acting on high tensor powers of the bundle exhibits gaps and clusters of eigenvalues. We prove that for each cluster the number of eigenvalues that it contains is given by a Riemann–Roch number. We also give a pointwise description of the Schwartz kernel of the spectral projectors onto the eigenstates of each cluster, similar to the Bergman kernel asymptotics of positive line bundles. Another result is that gaps and clusters also appear in local Weyl laws.

Keywords

Bochner Laplacian, magnetic Laplacian, Riemann–Roch number, Weyl law

Mathematical Subject Classification

Primary: 35P20, 35Q40, 58J20, 58J50

Milestones

Received: 12 October 2021

Revised: 2 November 2022

Accepted: 7 March 2023

Published: 19 July 2024

Authors

Laurent Charles
Institut de Mathématiques de Jussieu-Paris Rive Gauche Sorbonne Université, CNRS Paris France

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Vol. 17, No. 6, 2024