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Eigenvalue distribution in gaps of the essential spectrum of the Bochner-Schrödinger operator (2504.12928v1)

Published 17 Apr 2025 in math.SP, math-ph, math.DG, and math.MP

Abstract: The Bochner-Schr\"odinger operator $H_{p}=\frac 1p\Delta{Lp}+V$ on high tensor powers $Lp$ 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.

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