Spectral Asymptotics for Operators of Hormander Type

2013/10/31 by Andrew Ursitti, Andrew L. Ursitti, Ursitti, Andrew L.
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math.AP #math.SP

paper · pdf · doi:10.48550/arxiv.1310.8649

16 pages, submitted

arxiv created 2013/10/31 · openalex publication_date 2013/10/31 · arxiv updated 2013/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

An asymptotic equality of the form Tr_L² e^-t(L+V)=Ct^-α+o(t^-α) as t→ 0 is given for the trace of the heat semigroup generated by operators on compact manifolds of the form L+V=-∑_i=1^mX_i² +∑_i,j=1^mc_ij[X_i,X_j]+∑_i=1^m γ_iX_i+V for smooth real potentials (V) which satisfy Hörmander's bracket-generating condition. In the self-adjoint case, a Weyl law is proved for the spectra of such operators. Analogous results are proved for the Dirichlet boundary value problem.

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