Semiclassical asymptotic expansions for functions of the Bochner-Schrödinger operator

2022/05/18 by Kordyukov, Yuri A.
#Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2205.09011

Abstract

The Bochner-Schrödinger operator H_p=\frac 1pΔ^{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 is studied. For any function φ∈ \mathcal S(\mathbb R), we consider the bounded linear operator φ(H_p) in L²(X,L^p⊗ E) defined by the spectral theorem and describe an asymptotic expansion of its smooth Schwartz kernel in a fixed neighborhood of the diagonal in the semiclassical limit p→ ∞. In particular, we prove that the trace of the operator φ(H_p) admits a complete asymptotic expansion in powers of p^-1/2 as p→ ∞.

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