Electronic ISBN:  9781470401771 
Product Code:  MEMO/124/592.E 
List Price:  $40.00 
MAA Member Price:  $36.00 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 124; 1996; 58 ppMSC: Primary 58;
The subject of this memoir is the spectrum of a Diractype operator on an odddimensional 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 “eigenvalueLagrangians” 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 nonvanishing order). In certain cases, this leads to topological algorithms for computing spectral flow.ReadershipGraduate students and research mathematicians interested in global analysis and analysis on manifolds.

Table of Contents

Chapters

1. Introduction

2. Basics

3. Eigenvalue and tangential Lagrangians

4. Small extended $L^2$ eigenvalues

5. Dynamic properties of eigenvalue Lagrangians on $N^R_\lambda $ as $R \to \infty $

6. Properties of analytic deformations of extended $L^2$ eigenvalues

7. Time derivatives of extended $L^2$ and APS eigenvalues


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The subject of this memoir is the spectrum of a Diractype operator on an odddimensional 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 “eigenvalueLagrangians” 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 nonvanishing order). In certain cases, this leads to topological algorithms for computing spectral flow.
Graduate students and research mathematicians interested in global analysis and analysis on manifolds.

Chapters

1. Introduction

2. Basics

3. Eigenvalue and tangential Lagrangians

4. Small extended $L^2$ eigenvalues

5. Dynamic properties of eigenvalue Lagrangians on $N^R_\lambda $ as $R \to \infty $

6. Properties of analytic deformations of extended $L^2$ eigenvalues

7. Time derivatives of extended $L^2$ and APS eigenvalues