Spectral Asymptotics of Elliptic Operators on Manifolds
Ivan G. Avramidi

TL;DR
This paper reviews recent advances in spectral asymptotics of elliptic operators on manifolds, introducing new invariants and spectral functions that encode geometric information beyond traditional spectra.
Contribution
It introduces quantum heat traces, new invariants involving eigenfunctions, and algebraic methods for semigroup convolution, expanding the analysis of spectral properties of elliptic operators.
Findings
Development of quantum heat traces as spectral functions
Introduction of invariants depending on eigenfunctions
Algebraic computation of semigroup convolutions
Abstract
The study of spectral properties of natural geometric elliptic partial differential operators acting on smooth sections of vector bundles over Riemannian manifolds is a central theme in global analysis, differential geometry and mathematical physics. Instead of studying the spectrum of a differential operator directly one usually studies its spectral functions, that is, spectral traces of some functions of the operator, such as the spectral zeta function and the heat trace . The kernel of the heat semigroup , called the heat kernel, plays a major role in quantum field theory and quantum gravity, index theorems, non-commutative geometry, integrable systems and financial mathematics. We review some recent progress in the study of spectral asymptotics. We study more general spectral functions, such as ,…
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Taxonomy
Topicsadvanced mathematical theories · Geometric Analysis and Curvature Flows · Spectral Theory in Mathematical Physics
