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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 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 connected to its boundary.

The unifying thought at the back of the research of those varieties of spectra is the suggestion of sure "eigenvalue-Lagrangians" within the symplectic house $L^2(\partial M)$, an idea as a result of Mrowka and Nicolaescu. via learning the dynamics of those Lagrangians, the authors may be able to identify that these parts of the 2 varieties of spectra which go through 0 behave in primarily an identical means (to first non-vanishing order). every so often, this ends up in topological algorithms for computing spectral move.

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Let (^) f c , / ^ , etc. always denote the fcth derivative at 0. If i>o, ^o £ V^, choose t;*, v^ i = 1, • • • , n — 1 as in the definition of Vn. Then 0=< £(*)(£ ^ ) , ^ f i > - < ^Vit^DWfcvit*) t >. i i Taking nth derivatives and using the definition of Vn, we get < (^)B(Z>(t)(5>t<)),«o >=< vo,(£)nW)(Eeiti)) > • i i Thus Bn(vo,vo) is independent of the choice of Vi for i > 0, and hence J5n is well-defined. Since the inner product is Hermitian, so is Bn. It is trivial to check that V^+i C ker£?

3. Decompose Sk into the direct sum Sk = S£x 0 S^ A by taking s £\ = s P an ivk,\} and 5 s fc,A = 4. Let Pan iv-k,\}- W = kerA, -P\ = P\ = ®k>nSkX ®k>nSkfX (by which we mean the closure in L2(E) of this sum), so that L2(£) = PA-0W0PA+. 1) Notice that we have avoided the case A = /i* 7^ 0. We will chiefly be interested in small eigenvalues; in particular it suits our purpose to assume that |A| is smaller than the smallest positive eigenvalue /xn+i of the tangential operator A. In practice, we will consider small A.

Consider now an analytic 1-parameter family D(t),t G (—e, e) of Dirac operators with the same principal symbol in cylindrical form, as before. The next lemma shows that P^(t) vary analytically in t and A. 1 by working on L2(Y x (—oo, 0]). 1]. (We wish to thank U. 3 LEMMA. The subspaces PJ^it) vary analytically in t and A. Moreover, there exists an analytically varying family of (non-orthogonal) projections H\(t) : L2(E) -> P+(t) so that nx(t){avk,x(t) + bv-klX(t)) = avktX(t) for k> 0. Proof. 1 of [KK4] for details.

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