Exponential localization for eigensections of the Bochner-Schr\"odinger operator

TL;DR

This paper investigates the spectral behavior of the Bochner-Schr"odinger operator on high tensor powers of line bundles over manifolds with non-degenerate curvature, revealing exponential decay of eigensections in spectral gaps.

Contribution

It provides an asymptotic analysis of the spectrum of the operator, approximating it by model operators with constant magnetic fields, and establishes exponential decay of eigensections in spectral gaps.

Findings

Spectrum asymptotically matches model operators' spectra

Eigenfunctions decay exponentially outside compact sets

Spectral gaps contain discrete eigenvalues with localized eigensections

Abstract

We study asymptotic spectral properties of the Bochner-Schr\"odinger 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 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 Schr\"odinger 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 the compact subset.