Exponential Localization for Eigensections of the Bochner–Schrödinger operator

Abstract

We study asymptotic spectral properties of the Bochner–Schrödinger operator \(H_{p}=\frac 1p\Delta^{L^p\otimes E}+V\) 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 the assumption that the curvature form of \(L\) is nondegenerate. 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 \(p^{-1/4}\), 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 from a compact subset.

DOI 10.1134/S1061920824030099