HEAT KERNEL ASYMPTOTICS FOR ROOTS OF GENERALIZED LAPLACIANS

2003 ◽  
Vol 14 (04) ◽  
pp. 397-412 ◽  
Author(s):  
CHRISTIAN BÄR ◽  
SERGIU MOROIANU
Keyword(s):  
Heat Kernel ◽  
Closed Manifold ◽  
Type Operator ◽  
Wodzicki Residue ◽  
Heat Trace ◽  
Heat Kernel Asymptotics ◽  
Laplace Type

We describe the heat kernel asymptotics for roots of a Laplace type operator Δ on a closed manifold. A previously known relation between the Wodzicki residue of Δ and heat trace asymptotics is shown to hold pointwise for the corresponding densities.

scholarly journals HEAT KERNEL ASYMPTOTICS OF OPERATORS WITH NON-LAPLACE PRINCIPAL PART

2001 ◽  
Vol 13 (07) ◽  
pp. 847-890 ◽  
Author(s):  
IVAN G. AVRAMIDI ◽  
THOMAS BRANSON
Keyword(s):  
Asymptotic Expansion ◽  
Heat Kernel ◽  
Principal Part ◽  
Vector Bundles ◽  
Compact Riemannian Manifold ◽  
Differential Operators ◽  
Leading Order ◽  
Partial Differential Operators ◽  
Heat Kernel Asymptotics ◽  
Laplace Type

We consider second-order elliptic partial differential operators acting on sections of vector bundles over a compact Riemannian manifold without boundary, working without the assumption of Laplace-like principal part -∇μ∇μ. Our objective is to obtain information on the asymptotic expansions of the corresponding resolvent and the heat kernel. The heat kernel and the Green's function are constructed explicitly in the leading order. The first two coefficients of the heat kernel asymptotic expansion are computed explicitly. A new semi-classical ansatz as well as the complete recursion system for the heat kernel of non-Laplace type operators is constructed. Some particular cases are studied in more detail.

scholarly journals The Noncommutative Infinitesimal Equivariant Index Formula

2014 ◽  
Vol 14 (1) ◽  
pp. 73-102 ◽  
Author(s):  
Yong Wang
Keyword(s):  
Heat Kernel ◽  
Explicit Formula ◽  
Noncommutative Geometry ◽  
Index Formula ◽  
Equivariant Index ◽  
Heat Kernel Asymptotics

AbstractIn this paper, we establish an infinitesimal equivariant index formula in the noncommutative geometry framework using Greiner's approach to heat kernel asymptotics. An infinitesimal equivariant index formula for odd dimensional manifolds is also given. We define infinitesimal equivariant eta cochains, prove their regularity and give an explicit formula for them. We also establish an infinitesimal equivariant family index formula and introduce the infinitesimal equivariant eta forms as well as compare them with the equivariant eta forms.

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