# P. Kirk's Analytic Deformations of the Spectrum of a Family of Dirac PDF

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By P. Kirk

ISBN-10: 082180538X

ISBN-13: 9780821805381

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

The unifying thought in the back of the research of those kinds of spectra is the idea of definite "eigenvalue-Lagrangians" within the symplectic house $L^2(\partial M)$, an concept because of Mrowka and Nicolaescu. through learning the dynamics of those Lagrangians, the authors may be able to identify that these parts of the 2 kinds of spectra which go through 0 behave in basically a similar manner (to first non-vanishing order). often times, this results in topological algorithms for computing spectral movement.

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

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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1. The space U x C contains two analytic subvarieties, B and V(L(t)), defined as follows. The point ((£, A), M) eU x £ lies in B if the intersection of M with •P\~(t) is non-zero, and it lies in V(L(t)) if the intersection with L(t) 0 -P\"(t) is non-zero. Notice that B C V(L(t)) for any path L(t), and that the discontinuities of K lie along S = N~l(B). Indeed N~1(B) is exactly the union of the graphs of the type 1 eigenvalues, and N~l(V(L(t)) is the union of the graphs of all the eigenvalues. Moreover, B is the intersection of all the V(L(t)) over all choices of L(t).

Write L(0) 0P O + (O) as an orthogonal sum V 0 W. Then L2(E) = (V 0 JV) 0 (W®JW). Since M 0 (0)nTV = 0 and the M\(t) are commensurate, M\(t)C\W = 0 for (A,£) near (0,0). Consider the symplectic reduction p : £ —• £(V 0 JV) defined by Mx(t) ~ P(MA(t)) = AU M ;(t)nwr - c v 0 JV. Since MA(£) fl W = 0 for (A,£) near (0,0) the p(M\{t)) are an analytic family of finite dimensional Lagrangians in the symplectic space V 0 JV for (A, t) near (0,0). The construction is set up so that dim(p(Mx(t)) HV)= dim(Mx(t) n (L(0) 0 P 0 + ( 0 ))- Since p(Mo(0)) = V, the nearby Lagrangians p(M\(t)) are transverse to J V , hence p(Mx(t)) = {m + JZ(A, t)m |m G V} for some analytic family of self-adjoint linear maps Z(\,t) : V —> V.

Then Ux(t) = \{Id — Zx(t)) — proj^ t is analytic in ANALYTIC DEFORMATIONS 23 t and A and is a projection to Px(t) satisfying the conditions of the lemma. A similar argument applies to Px(t)| Remark. Prom the proof, one can easily see what modifications need to be made to take into account the case when H(t) jumps up in dimension, namely, near t0 "redefine" H(t) to be the span of eigenvectors whose analytic extension to the interval pass through 0 at t = toThe -PjfOO are not Lagrangians if H(t) is non-zero, since H(t) is orthogonal to both -P\(£) and J(P^(t)).