Analytic Deformations of the Spectrum of a Family of Dirac by P. Kirk

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, relatively, how this spectrum varies less than an analytic perturbation of the operator. sorts 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 connected to its boundary.

The unifying notion in the back of the research of those sorts of spectra is the thought of convinced "eigenvalue-Lagrangians" within the symplectic area $L^2(\partial M)$, an idea because of Mrowka and Nicolaescu. via learning the dynamics of those Lagrangians, the authors may be able to determine that these parts of the 2 varieties of spectra which go through 0 behave in basically an identical manner (to first non-vanishing order). in some cases, this ends up in topological algorithms for computing spectral circulation.

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If, moreover, K is continuous at (0,0), then L(t) is transverse to K(t,\) for small (£,A) and so N*(t)(t) n (L(t) e P+(t)(t)) = N*(t)(t) n P+(t)(t). 7 again this implies that the kernel of D(0) consists only of type 1 eigenvectors. If Xi(t) is an extended I? 7 again one sees that L(t) fl K(t, Xi(t)) is not zero for t ^ 0, and in the case of a type 3 eigenvalue, 1,(0) D K(0,0) is not zero. 3 . 1 is obvious: first separate the (f>i{t) into those that are L2 for all t, those that are I? only at t = 0, and the rest.

2. The form Bn : Vn x Vn —> C satisfies ^ n f e ( 0 ) , ^ ( 0 ) ) = - < t*(0), — Aj n ) ^(0) > ^A|(0)0) J and Bn(vmM*)) = -< *i(0), f ? , dX\{0t0) n) A} 6,(0) > . 1 will be completed by showing that dT — :V ->V ^ A |(o,o) and are positive definite. To see that |gjj . 4), &X | (0)0) is positive definite, Let v,vf v+ e V. Since G(0) = Id JT(X,0)vep(Nx(0)). Examining the definition of symplectic reduction one sees that there is a path w(X) in W with w(0) = 0 such that v + 7T(A, 0)v + 10(A) G iVA(0).

This dependence can be illustrated in geometric terms. For example, suppose that X(t) is any real-analytic function with A(0) = 0. 9 shows that the family K(t, X(t)) is analytic for small t not equal to zero, and can be extended to an analytic path L(t) of Lagrangians through t = 0. By construction, one immediately sees that X(t) is an extended L2 eigenvalue which respects (L(t),R), with multiplicity at least \ dimW(O). In fact for this choice of L(t) there are at least \ dim H(0) linearly independent type 2 or 3 kernel elements at t = 0.

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