Semiclassical trace formula for the Bochner-Schrödinger operator

2025/02/17 by Yuri A. Kordyukov, Kordyukov, Yuri A.
Mathematics · #Advanced Mathematical Physics Problems #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2502.12087

openalex publication_date 2025/02/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the semiclassical Bochner-Schrödinger operator H_p=(1)/(p²)Δ^{Lp⊗ E}+V on tensor powers L^p of a Hermitian line bundle L twisted by a Hermitian vector bundle E on a Riemannian manifold of bounded geometry. For any function φ∈ C^∞_c(\mathbb R), we consider the bounded linear operator φ(H_p) in L²(X,L^p⊗ E) defined by the spectral theorem. We prove that its smooth Schwartz kernel on the diagonal admits a complete asymptotic expansion in powers of p^-1 in the semiclassical limit p→ ∞. In particular, when the manifold is compact, we get a complete asymptotic expansion for the trace of φ(H_p).

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