Eigenvalue distribution in gaps of the essential spectrum of the Bochner-Schrödinger operator

The Bochner-Schrödinger operator $H_p=\frac{1}{p}{\mathrm{\Delta }}^{L^p}+V$ 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 get a Weyl type asymptotic formula for the eigenvalue counting function.