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Dirac operators and spectral geometry

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Published by Cambridge University Press in Cambridge, New York .
Written in English


  • Spectral geometry.,
  • Differential operators.,
  • Mathematical physics.

Book details:

Edition Notes

Includes bibliographical references (p. 191-206) and index.

StatementGiampiero Esposito.
SeriesCambridge lecture notes in physics ;, 12
LC ClassificationsQC20.7.S64 E78 1998
The Physical Object
Paginationxiii, 209 p. ;
Number of Pages209
ID Numbers
Open LibraryOL377471M
ISBN 100521648629
LC Control Number98039477

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1. The Dirac operator; 2. Differential operators on manifolds; 3. Index problems; 4. Spectral asymmetry; 5. Spectral geometry with operators of Laplace type; 6. New Cited by: The Dirac operator has many useful applications in theoretical physics and mathematics. This book provides a clear, concise and self-contained introduction to the global theory of the Dirac operator and to the analysis of spectral asymptotics with local or nonlocal boundary conditions. D/ is an (unbounded) operator on a Hilbert space H = L2 (M, S) of “square-integrable spinors” and C ∞ (M) also acts on H by multiplication / f ]k = k grad f k∞. operators with k [D, Noncommutative geometry generalizes (C ∞ (M), L2 (M, S), D) / to a spectral . Noncommutative geometry generalizes (C∞(M),L2(M,S),D/) to a spectral triple of the form (A,H,D), where Ais a “smooth” algebra acting on a Hilbert space H, Dis an (unbounded) selfadjoint operator on H, subject to certain conditions: in particular that [D,a] be a bounded operator for each a∈ by: 6.

  The eigenvalues of the Laplacian are naturally linked to the geometry of the manifold. For manifolds that admit spin (or) structures, one obtains further information from equations involving Dirac operators and spinor fields. In the case of four-manifolds, for example, one has the remarkable Seiberg-Witten. DIRAC OPERATORS AND SPECTRAL TRIPLES FOR SOME FRACTALS 3 Hausdorffmeasure canberecovered fromoperatoralgebraicdata. Work in this direction was pursued by Daniele Guido and Tommaso Isola in several papers, [16, 17, 18]. Earlier, in [27], using the results and methods of [21] and [6] (includ-. Elliptic operators self-adjoint extensions Sobolev spaces p-Laplacian spectral theory Schauder estimates regularity theory Dirac operator Authors and affiliations D. E. Edmunds. We construct spectral triples and, in particular, Dirac operators, for the algebra of continuous functions on certain compact metric spaces. The triples are countable sums of triples where each summand is based on a curve in the space.

Riemannian geometry in completely algebraic terms, using a -algebra A represented on a Hilbert space H (which capture the topological aspects), and a not necessarily bounded generalized Dirac operator D(which captures the metric aspects). These elements form a spectral triple and are at the basis of the construction. These ingredients are naturallyCited by: Asymptotic Formulae in Spectral Geometry Peter B. Gilkey A great deal of progress has been made recently in the field of asymptotic formulas that arise in the theory of Dirac and Laplace type operators. Provides a clear, concise and self-contained introduction to the global theory of the Dirac operator and to the analysis of spectral asymptotics with local or non-local boundary conditions. Ideal for graduate students and researchers working in theoretical physics and mathematics. The basics of different ingredients will be presented and studied like, Dirac operators, heat equation asymptotics, zeta functions and then, how to get within the framework of operators Author: Oussama Hijazi.