Perturbation Determinant, Krein's Shift Function and Index Theorem

2009 ◽  
pp. 157-171
Author(s):  
Kalyan B. Sinha
Keyword(s):  
Index Theorem ◽  
Shift Function ◽  
Perturbation Determinant

Some Applications of the Perturbation Determinant in Finite von Neumann Algebras

2010 ◽  
Vol 62 (1) ◽  
pp. 133-156 ◽  
Author(s):  
Konstantin A. Makarov ◽  
Anna Skripka
Keyword(s):  
Von Neumann Algebras ◽  
Spectral Shift ◽  
Linear Perturbation ◽  
Shift Function ◽  
Differentiation Formula ◽  
Von Neumann ◽  
Finite Von Neumann Algebra ◽  
Finite Von Neumann Algebras ◽  
Perturbation Determinant ◽  
Spectral Averaging

AbstractIn the finite von Neumann algebra setting, we introduce the concept of a perturbation determinant associated with a pair of self-adjoint elements H0 and H in the algebra and relate it to the concept of the de la Harpe-Skandalis homotopy invariant determinant associated with piecewise C1-paths of operators joining H0 and H. We obtain an analog of Krein's formula that relates the perturbation determinant and the spectral shift function and, based on this relation, we derive subsequently (i) the Birman-Solomyak formula for a general non-linear perturbation, (ii) a universality of a spectral averaging, and (iii) a generalization of the Dixmier-Fuglede-Kadison differentiation formula.

scholarly journals Index theorem on T2/ZN orbifolds

2021 ◽  
Vol 103 (2) ◽  
Author(s):  
Makoto Sakamoto ◽  
Maki Takeuchi ◽  
Yoshiyuki Tatsuta
Keyword(s):  
Index Theorem

scholarly journals An index theorem for super derivations

1989 ◽  
Vol 125 (1) ◽  
pp. 147-152
Author(s):  
Arthur Jaffe ◽  
Andrzej Lesniewski
Keyword(s):  
Index Theorem

Bundles over Quantum Sphere and Noncommutative Index Theorem

K-Theory ◽  
2000 ◽  
Vol 21 (2) ◽  
pp. 141-150 ◽  
Author(s):  
Piotr M. Hajac
Keyword(s):  
Index Theorem ◽  
Quantum Sphere

A local index theorem for non K�hler manifolds

1989 ◽  
Vol 284 (4) ◽  
pp. 681-699 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Index Theorem ◽  
Local Index ◽  
Local Index Theorem

scholarly journals Periodic Solutions and Nontrivial Periodic Solutions for a Class of Rayleigh-Type Equation with Two Deviating Arguments

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Meiqiang Feng
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Periodic Solutions ◽  
Point Theorem ◽  
Type Equation ◽  
Index Theorem ◽  
Rayleigh Equation ◽  
Deviating Arguments ◽  
Nontrivial Periodic Solutions ◽  
Existence Of Periodic Solutions

The Rayleigh equation with two deviating argumentsx′′(t)+f(x'(t))+g1(t,x(t-τ1(t)))+g2(t,x(t-τ2(t)))=e(t)is studied. By using Leray-Schauder index theorem and Leray-Schauder fixed point theorem, we obtain some new results on the existence of periodic solutions, especially for the existence of nontrivial periodic solutions to this equation. The results are illustrated with two examples, which cannot be handled using the existing results.

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