The semiclassical structure of low-energy states in the presence of a magnetic field

Abstract

We consider a compact Riemannian manifold with a Hermitian line bundle whose curvature is non-degenerate. The Laplacian acting on high tensor powers (the semiclassical regime) of the bundle exhibits a cluster of low-energy states. We demonstrate that the orthogonal projectors onto these states are the Fourier components of an operator with the structure of the Szeg\"o projector, i.e. a Fourier integral operator of Hermite type. This result yields semiclassical asymptotics for the low-energy eigenstates.