Exponential localization for eigensections of the Bochner-Schrödinger operator

We study asymptotic spectral properties of the Bochner-Schrödinger operator $H_p=\frac{1}{p}{\mathrm{\Delta }}^{L^p\otimes E}+V$ on high tensor powers of a Hermitian line bundle $L$ twisted by a Hermitian vector bundle $E$ on a Riemannian manifold $X$ of bounded geometry under assumption that the curvature form of $L$ is non-degenerate. At an arbitrary point $x_0$ of $X$ the operator $H_p$ can be approximated by a model operator ${\mathcal{H}}^{(x_0)}$, which is a Schrödinger operator with constant magnetic field. For large $p$, the spectrum of $H_p$ asymptotically coincides, up to order $p^{-1\mathrm{/}4}$, with the union of the spectra of the model operators ${\mathcal{H}}^{(x_0)}$ over $X$. We show that, if the union of the spectra of ${\mathcal{H}}^{(x_0)}$ over the complement of a compact subset of $X$ has a gap, then the spectrum of $H_p$ in the gap is discrete and the corresponding eigensections decay exponentially away the compact subset.