# Analytic Deformations of the Spectrum of a Family of Dirac Operators on an Odd-Dimensional Manifold with Boundary

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*P. Kirk; E. Klassen*

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.

#### Table of Contents

# Table of Contents

## Analytic Deformations of the Spectrum of a Family of Dirac Operators on an Odd-Dimensional Manifold with Boundary

- Contents vii8 free
- Chapter 1. Introduction 110 free
- Chapter 2. Basics 716 free
- Chapter 3. Eigenvalue and tangential Lagrangians 1827
- Chapter 4. Small extended L[sup(2)] eigenvalues 2837
- Chapter 5. Dynamic properties of eigenvalue Lagrangians on N[sup(R)sub(λ)]; as R → ∞ 3746
- Chapter 6. Properties of analytic deformations of extended L[sup(2)] eigenvalues 4049
- 6.1. The three types of extended L[sup(2)] eigenvectors 4049
- 6.2. The effect of the different choices of L[sup(2)] on the eigenvalues and the non–stability of L[sup(2)]eigenvalues 4251
- 6.3. Derivatives of extended L[sup(2)] eigenvectors 4352
- 6.4. The Hermitian forms controlling the deformations of extended L[sup(2)] eigenvalues have signature independent of R 4655

- Chapter 7. Time derivatives of extended L[sup(2)] and APS eigenvalues 5059
- Bibliography 5766