The canonical integral and noncommutative residue on symbols

2012 ◽  
pp. 15-34
Author(s):  
Sylvie Paycha
Keyword(s):  
Noncommutative Residue ◽  
Canonical Integral

scholarly journals Dirac Functional Determinants in Terms of the Eta Invariant and the Noncommutative Residue

2003 ◽  
Vol 237 (3) ◽  
pp. 507-532 ◽  
Author(s):  
Guido Cognola ◽  
Emilio Elizalde ◽  
Sergio Zerbini
Keyword(s):  
Eta Invariant ◽  
Noncommutative Residue

The Noncommutative Residue about Witten Deformation

2018 ◽  
Vol 35 (4) ◽  
pp. 550-568
Author(s):  
Kai Hua Bao ◽  
Ai Hui Sun ◽  
Chao Deng
Keyword(s):  
Noncommutative Residue ◽  
Witten Deformation

scholarly journals Homology and residues of adiabatic pseudodifferential operators

2004 ◽  
Vol 175 ◽  
pp. 171-221 ◽  
Author(s):  
Sergiu Moroianu
Keyword(s):  
Existence And Uniqueness ◽  
Pseudodifferential Operators ◽  
Hochschild Homology ◽  
Total Space ◽  
Homology Groups ◽  
Noncommutative Residue

AbstractWe compute the Hochschild homology groups of the adiabatic algebra Ψa(X), a deformation of the algebra of pseudodifferential operators Ψ(X) when X is the total space of a fibration of closed manifolds. We deduce the existence and uniqueness of traces on Ψa(X) and some of its ideals and quotients, in the spirit of the noncommutative residue of Wodzicki and Guillemin. We introduce certain higher homological versions of the residue trace. When the base of the fibration is S1, these functionals are related to the η function of Atiyah-Patodi-Singer.

scholarly journals The Noncommutative Residue for Manifolds with Boundary

1996 ◽  
Vol 142 (1) ◽  
pp. 1-31 ◽  
Author(s):  
Boris V. Fedosov ◽  
François Golse ◽  
Eric Leichtnam ◽  
Elmar Schrohe
Keyword(s):  
Manifolds With Boundary ◽  
Noncommutative Residue

scholarly journals ON THE SINGULARITIES OF THE ZETA AND ETA FUNCTIONS OF AN ELLIPTIC OPERATOR

2012 ◽  
Vol 23 (06) ◽  
pp. 1250020
Author(s):  
PAUL LOYA ◽  
SERGIU MOROIANU ◽  
RAPHAËL PONGE
Keyword(s):  
Elliptic Operator ◽  
Differential Operators ◽  
Pseudodifferential Operators ◽  
Zeta Functions ◽  
Dirac Type ◽  
Noncommutative Residue ◽  
Analytical Proof ◽  
Spectral Asymmetry ◽  
Operator Of Dirac Type ◽  
Funct Anal

Let P be a self-adjoint elliptic operator of order m > 0 acting on the sections of a Hermitian vector bundle over a compact Riemannian manifold of dimension n. General arguments show that its zeta and eta functions may have poles only at points of the form [Formula: see text], where k ranges over all nonzero integers ≤ n. In this paper, we construct elementary and explicit examples of perturbations of P which make the zeta and eta functions become singular at all points at which they are allowed to have singularities. We proceed within three classes of operators: Dirac-type operators, self-adjoint first-order differential operators and self-adjoint elliptic pseudodifferential operators. As consequences, we obtain genericity results for the singularities of the zeta and eta functions in those settings. In particular, in the setting of Dirac-type operators we obtain a purely analytical proof of a well-known result of Branson–Gilkey [Residues of the eta function for an operator of Dirac type, J. Funct. Anal. 108(1) (1992) 47–87], which was obtained by invoking Riemannian invariant theory. As it turns out, the results of this paper contradict Theorem 6.3 of [R. Ponge, Spectral asymmetry, zeta functions and the noncommutative residue, Int. J. Math. 17 (2006) 1065–1090]. Corrections to that statement are given in this paper.

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