By P. Kirk

The topic of this memoir is the spectrum of a Dirac-type operator on an odd-dimensional manifold M with boundary and, fairly, how this spectrum varies below an analytic perturbation of the operator. forms of eigenfunctions are thought of: first, these pleasurable the "global boundary stipulations" of Atiyah, Patodi, and Singer and moment, these which expand to $L^2$ eigenfunctions on M with an enormous collar hooked up to its boundary.

The unifying inspiration at the back of the research of those different types of spectra is the idea of convinced "eigenvalue-Lagrangians" within the symplectic area $L^2(\partial M)$, an concept because of Mrowka and Nicolaescu. by means of learning the dynamics of those Lagrangians, the authors may be able to determine that these parts of the 2 kinds of spectra which go through 0 behave in basically an analogous means (to first non-vanishing order). now and again, this results in topological algorithms for computing spectral movement.

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**Extra resources for Analytic Deformations of the Spectrum of a Family of Dirac Operators on an Odd-Dimensional Manifold With Boundary**

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The limit: Zx(t)= lim - / (A(t) + J\-T)~ldT n-KX) 7T JYn exists pointwise. The difference Zo(0) — Z\{t) is a norm-convergent integral (in any L2(E) norm), and using Neumann series one sees that this difference is analytic in t and A. Notice that A(t) + JX — r decomposes as a block sum according to the decomposition L2(E) = ®Sk(t). Moreover, on Sk(t), in the basis ij)k(t)^^k{i), Inverting this matrix and integrating along Fn one computes that t -avkix(t) - bv-k,x(t) if 0 < fc < n. Thus | ( J d + ^ A ( ^ ) ) is analytic in £ and A and is the projection to Px(t) 0 W(£) corresponding to the (non-orthogonal) direct sum L2(E) = Px(t) ®H(t)Px(t).

Lagrangians in H are used to define self-adjoint Atiyah-Patodi-Singer boundary conditions. Given a Lagrangian L C 7i, the operator £>: C°°(£; L 0 P0+) - • C°°(£) is self adjoint, and Predholm. 5 DEFINITION. Call eigenvalues and eigenvectors of D with L 0 P o + boundary values Atiyah-Patodi-Singer (APS) eigenvalues and eigenvectors which respect L. More generally, eigenvectors and eigenvalues of D on X(R) with L0P^" boundary conditions are called Atiyah-Patodi-Singer eigenvalues and eigenvectors which respect the pair (L,R).

2) exponentially decays in u. Direct calculations show that (^)r(e-(u-fl)vW*)2-M*)2) dt = (ilL)r(e-(ii-fl)/ifc(i)) dt for v < m (in fact for r < 2m) and (±v( (dt) ^_/^)r^W ("WWW)- | (jLrv,fc(f) + A(m)_iMV,_fc(0) ifr