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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 below an analytic perturbation of the operator. kinds 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 unlimited collar connected to its boundary.

The unifying inspiration at the back of the research of those forms of spectra is the suggestion of yes "eigenvalue-Lagrangians" within the symplectic area $L^2(\partial M)$, 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 sorts of spectra which go through 0 behave in basically an identical means (to first non-vanishing order). on occasion, this results in topological algorithms for computing spectral move.

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1. It will be convenient below to express this family as a composite of two families; one which controls the t dependence and one which controls the A dependence. 3 and the projections to L(t) to construct an analytic family of symplectomorphisms F(t) : L2(E) -+ L2(E) which satisfies F(t)(L(t0) © P 0 + (*O)) = L(t) e p 0 + (t). 1 corresponding to the A parameter) to obtain an analytic family $(\,t):L2(E)^L2(E) of symplectomorphisms which satisfy: $(A,()(i(t)eP 0 + W) = Ht) e pA+(t). We state the properties of these maps in the following proposition.

Wxt + --- + ^mtm) for all k < m. 7 Vo) = < (jt)m(D(t)(v0 + « ! Vo THEOREM. (i) The form Bm is finite, well-defined, and Hermitian. >*(oo) . The kernel of Bm is Vrn+l- (ii) Jy{0i(£)}£ = i is a collection of extended L2 eigenvectors of D(t) with eigenvalues Xi(t) which respect (L(t),R) and so that W = span{(f)i(ti)}, then Vm = span{(j)i(Q) I \f] =0 fork where A m : Vm —> Vm is defined by Am<^(0) = A^ Vi(0)- In particular, the signature and kernel of A m are the same as the signature and kernel of Bm and hence are independent of R.

The eigenvalue Lagrangians and the negative tangential Lagrangians are contained in the space £ of Lagrangians commensurate to Po~(0). 3 Extended L2 eigenvectors on X(oo) The Pjjk are important because they can be used to decide when a A eigenvector on X(0) extends to an L2 or "extended" L2 eigenvector on X(oo). 7 PROPOSITION. Let u e JCi/2(D - A), so u is a A eigenvector of D and has a restriction r(uj) £ N\. Assume either that X ^ fj,k for all k or else that A = 0. Then: 1. UJ extends to an L2 X-eigenvector of D on X(oo) if and only if r{uj) lies in ®LLk>\\\Sk,\ In particular, if \X\ < /in+i> then u extends to an L2 X-eigenvector on X(oo) if and only ifr(u) G P^".

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