On the spectrum of nondegenerate magnetic Laplacians

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.