## Download e-book for iPad: Analytic Deformations of the Spectrum of a Family of Dirac by P. Kirk

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. different types of eigenfunctions are thought of: first, these pleasing 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 different types of spectra is the idea of definite "eigenvalue-Lagrangians" within the symplectic area $L^2(\partial M)$, an concept as a result of Mrowka and Nicolaescu. by way of learning the dynamics of those Lagrangians, the authors may be able to identify that these parts of the 2 different types of spectra which go through 0 behave in basically an analogous means (to first non-vanishing order). occasionally, this results in topological algorithms for computing spectral stream.

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

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To see that B is negative definite, first observe that since W\0 — Id. 1) aA|A0 aA|A0 We first show that the second term is strictly negative. 1. Thus yx = r(Y\), the restriction to the boundary, and Y\ is a A-eigenvector of D. 2). dA\x0 aA|A0 = A0 < YXo, -7T dX |Ao aA|A0 Yx>-< YXo,YXo >-< = -lin„n2kvkyXo' k>n The spaces Sk are orthogonal and preserved by J , and so it suffices to show that ANALYTIC DEFORMATIONS 31 and are both non-positive.

Such a basis is easily found. Moreover, if a (finite-dimensional) Lagrangian subspace of kevA is specified one can choose the fa, k = 1, • • • ,n to span this subspace. Remark. /i n +i is the smallest positive eigenvalue of the tangential operator A. This number will appear frequently below. We next introduce various subspaces of L2(E) defined in terms of the tangential operator A. Fix a branch of the square root with y/r > 0 and \f^r — iy/r for r > 0. Let A be a real number such that either A ^ Hk for any fc, or A = 0.

Hence, v-f JT(X,0)v + w(X) e JVA(0) = J7(A, O)x(A) for some path x(A) in N0(0). Since 17(0,0) = Id, x(0) = v. Differentiating with respect to A at A = 0 and taking the inner product with Jv', one obtains: < J—v, J v ' > 4- < -^r, J*/ > = < ^rv, Jvf > + < ^ - , J v ' > . 3) dA| {00) dA c/A |(O|0) dA P. KIRK AND E. KLASSEN 56 Since ^ is in W, it is orthogonal to Jv'. 3) reduces to: Similarly | j is orthogonal to Jv'. 4 now implies that ^ , A similar argument shows that dS , 7 , is positive definite.

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### Analytic Deformations of the Spectrum of a Family of Dirac Operators on an Odd-Dimensional Manifold With Boundary by P. Kirk

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