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Exponential localization for eigensections of the Bochner-Schrödinger operator (2404.19684v1)

Published 30 Apr 2024 in math.SP, math-ph, math.DG, and math.MP

Abstract: We study asymptotic spectral properties of the Bochner-Schr\"odinger operator $H_{p}=\frac 1p\Delta{Lp\otimes E}+V$ 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.

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