Abstract
The Bochner–Schrödinger operator \(H_{p}=\frac 1p\Delta^{L^p}+V\) on high tensor powers \(L^p\) of a Hermitian line bundle \(L\) on a Riemannian manifold \(X\) of bounded geometry is studied under the assumption of non-degeneracy of the curvature form of \(L\) . For large \(p\) , the spectrum of \(H_p\) asymptotically coincides with the union of all local Landau levels of the operator at the points of \(X\) . Moreover, if the union of the local Landau levels over the complement of a compact subset of \(X\) has a gap, then the spectrum of \(H_{p}\) in the gap is discrete. The main result of the paper is the trace asymptotics formula associated with these eigenvalues. As a consequence, we obtain a Weyl type asymptotic formula for the eigenvalue counting function.
