Eigenvalue distribution in gaps of the essential spectrum of the Bochner-Schrödinger operator
Yuri A. Kordyukov
TL;DR
This work analyzes the Bochner-Schrödinger operator $H_p=\frac{1}{p}\Delta^{L^p}+V$ on high tensor powers of a Hermitian line bundle over a manifold of bounded geometry with nondegenerate curvature. In the semiclassical limit $p\to\infty$, the spectrum concentrates near local Landau levels $\Lambda_{\mathbf k}(x)$ determined by the pointwise magnetic field; gaps in the essential spectrum yield discrete spectra in those gaps. The authors prove a detailed trace expansion $\operatorname{tr}\varphi(H_p) \sim p^{n}\sum_{r=0}^{\infty} \langle f_r,\varphi\rangle p^{-r/2}$ with an explicit leading coefficient $\langle f_0,\varphi\rangle=\frac{1}{(2\pi)^n}\sum_{\mathbf k} \int_X \varphi(\Lambda_{\mathbf k}(x)) d\mu(x)$, where $d\mu$ is the Liouville measure on $(X,\mathbf B)$. This yields a Weyl-type eigenvalue counting formula in the gaps and extends known results for magnetic Schrödinger operators to the geometric, bounded-geometry setting. The approach combines a local trace formula with global localization and semiclassical estimates to obtain precise asymptotics and a robust spectral gap analysis.
Abstract
The Bochner-Schrödinger operator $H_{p}=\frac 1pΔ^{L^p}+V$ 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.