We found a match
Your institution may have access to this item. Find your institution then sign in to continue.
- Title
Spectral asymptotics of elliptic operators on manifolds.
- Authors
Avramidi, Ivan G.
- 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 L directly one usually studies its spectral functions, that is, spectral traces of some functions of the operator, such as the spectral zeta function ζ (s) = Tr L − s and the heat trace Θ (t) = Tr exp (− t L). The kernel U (t ; x , x ′) of the heat semigroup exp (− t L) , 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 Tr f (t L) , that we call quantum heat traces. Also, we define new invariants of differential operators that depend not only on the eigenvalues but also on the eigenfunctions, and, therefore, contain much more information about the geometry of the manifold. Furthermore, we study some new invariants, such as Tr exp (− t L +) exp (− s L −) , that contain relative spectral information of two differential operators. Finally, we show how the convolution of the semigroups of two different operators can be computed by using purely algebraic methods.
- Subjects
DIFFERENTIAL invariants; ELLIPTIC operators; PARTIAL differential operators; QUANTUM field theory; MATHEMATICAL physics; DIFFERENTIAL operators; ZETA functions
- Publication
Reviews in Mathematical Physics, 2024, Vol 36, Issue 5, p1
- ISSN
0129-055X
- Publication type
Article
- DOI
10.1142/S0129055X24500077