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 DIRAC OPERATOR IN MATRIX GEOMETRY

2005 ◽  
Vol 02 (02) ◽  
pp. 227-264 ◽  
Author(s):  
IVAN G. AVRAMIDI
Keyword(s):  
Dirac Operator ◽  
Heat Kernel ◽  
Linear Combination ◽  
Differential Operators ◽  
Spectral Invariants ◽  
First Order ◽  
Partial Differential Operators ◽  
Laplace Type ◽  
Hilbert Action

We review the construction of the Dirac operator and its properties in Riemannian geometry, and show how the asymptotic expansion of the trace of the heat kernel determines the spectral invariants of the Dirac operator and its index. We also point out that the Einstein–Hilbert functional can be obtained as a linear combination of the first two spectral invariants of the Dirac operator. Next, we report on our previous attempts to generalize the notion of the Dirac operator to the case of Matrix Geometry, where, instead of a Riemannian metric there is a matrix valued self-adjoint symmetric two-tensor that plays a role of a "non-commutative" metric. We construct invariant first-order and second-order self-adjoint elliptic partial differential operators, which can be called "non-commutative" Dirac operators and non-commutative Laplace operators. We construct the corresponding heat kernel for the non-commutative Laplace type operator and compute its first two spectral invariants. A linear combination of these two spectral invariants gives a functional that can be considered as a non-commutative generalization of the Einstein–Hilbert action.

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.

A LAPLACE INTEGRAL, THE T–Y–Z EXPANSION, AND BEREZIN'S TRANSFORM ON A KÄHLER MANIFOLD

2005 ◽  
Vol 02 (03) ◽  
pp. 359-371 ◽  
Author(s):  
ANDREA LOI
Keyword(s):  
Asymptotic Expansion ◽  
Complex Manifold ◽  
Lower Order ◽  
Differential Operators ◽  
Expansion Formula ◽  
Covariant Derivatives ◽  
Open Set ◽  
Type Integral ◽  
Lower Order Terms ◽  
Laplace Type

Let M be an n-dimensional complex manifold endowed with a C∞ Kähler metric g. We show that a certain Laplace-type integral ℒm(x), when x varies in a sufficiently small open set U ⊂ M, has an asymptotic expansion [Formula: see text], where Cr:C∞(U) → C∞(U) are smooth differential operators depending on the curvature of g and its covariant derivatives. As a consequence we furnish a different proof of Lu's theorem by computing the lower order terms of Tian–Yau–Zelditch expansion in terms of the operator Cj. Finally, we compute the differential operators Qj of the expansion Ber m(f) = Σr≥0m-rQr(f) of Berezin's transform in terms of the operators Cj.

scholarly journals Jets and differential linear logic

2020 ◽  
pp. 1-27
Author(s):  
James Wallbridge
Keyword(s):  
Vector Bundles ◽  
Differential Operators ◽  
Linear Logic ◽  
Variational Calculus ◽  
Functional Derivative ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
Kleisli Category ◽  
New Interpretation ◽  
Linear Differential

Abstract We prove that the category of vector bundles over a fixed smooth manifold and its corresponding category of convenient modules are models for intuitionistic differential linear logic. The exponential modality is modelled by composing the jet comonad, whose Kleisli category has linear differential operators as morphisms, with the more familiar distributional comonad, whose Kleisli category has smooth maps as morphisms. Combining the two comonads gives a new interpretation of the semantics of differential linear logic where the Kleisli morphisms are smooth local functionals, or equivalently, smooth partial differential operators, and the codereliction map induces the functional derivative. This points towards a logic, and hence a computational theory of non-linear partial differential equations and their solutions based on variational calculus.

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