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Analytic Deformations of the Spectrum of a Family of Dirac Operators on an Odd-Dimensional Manifold with Boundary
P. Kirk, Indiana University, Bloomington, IN, and E. Klassen, Florida %State University, Tallahassee, FL
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Memoirs of the American Mathematical Society
1996; 58 pp; softcover
Volume: 124
ISBN-10: 0-8218-0538-X
ISBN-13: 978-0-8218-0538-1
List Price: US$38
Individual Members: US$22.80
Institutional Members: US$30.40
Order Code: MEMO/124/592
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The subject of this memoir is the spectrum of a Dirac-type operator on an odd-dimensional manifold M with boundary and, particularly, how this spectrum varies under an analytic perturbation of the operator. Two types of eigenfunctions are considered: first, those satisfying the "global boundary conditions" of Atiyah, Patodi, and Singer and second, those which extend to \(L^2\) eigenfunctions on M with an infinite collar attached to its boundary.

The unifying idea behind the analysis of these two types of spectra is the notion of certain "eigenvalue-Lagrangians" in the symplectic space \(L^2(\partial M)\), an idea due to Mrowka and Nicolaescu. By studying the dynamics of these Lagrangians, the authors are able to establish that those portions of the two types of spectra which pass through zero behave in essentially the same way (to first non-vanishing order). In certain cases, this leads to topological algorithms for computing spectral flow.

Readership

Graduate students and research mathematicians interested in global analysis and analysis on manifolds.

Table of Contents

  • Introduction
  • Basics
  • Eigenvalue and tangential Lagrangians
  • Small extended \(L^2\)-eigenvalues
  • Dynamic properties of eigenvalue Lagrangians on \(N^R_\lambda\) as \(R\rightarrow \infty\)
  • Properties of analytic deformations of extended \(L^2\) eigenvalues
  • Time derivatives of extended \(L^2\) and APS eigenvalues
  • Bibliography
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