# 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, quite, how this spectrum varies below an analytic perturbation of the operator. forms of eigenfunctions are thought of: first, these gratifying 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 concept in the back of the research of those kinds of spectra is the concept of convinced "eigenvalue-Lagrangians" within the symplectic house $L^2(\partial M)$, an idea as a result of Mrowka and Nicolaescu. by means of learning the dynamics of those Lagrangians, the authors may be able to determine that these parts of the 2 kinds of spectra which go through 0 behave in basically a similar method (to first non-vanishing order). occasionally, this results in topological algorithms for computing spectral circulation.

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We suppress the parameter t which is fixed throughout this proof; thus $x =$(A,£o)-) Let Wx = Q^1 : L2(E) - • L2{E). Notice that N\CiL®P£ = WX(NX) n L 0 P j . Thus we must show that WA(JVA) H L 0 Pj* = 0 for A near A0, but different from A0 We define a Hermitian form B : V x V —• C as follows. Given x, y G V, choose any smooth family yx G L2(E), A in some neighborhood of Ao, so that 1. ) Then define We claim that B is well-defined, Hermitian, and negative definite. To see that B is negative definite, first observe that since W\0 — Id.

This same equation also shows that Bm is Hermitian. We prove (ii) by induction on m. Notice that (ii) implies that the kernel of Bm is Vm+i. For the first step in the induction, suppose that vo — 0i(O). Then set v\ = 0^(0), and a(t) = proj W ( t )0i(t). 4 one concludes that 0 o € Vu so that Vi = W. One computes fli(fc(0),^(0)) = < ft(D(t)Mt))Aj(0) > = =. i(i>,i/;) = < Ait;,it; > for any v,w G V\. It follows that the signature and kernel of B\ equals that of Ai. In particular, the signature and kernel of Ai, which a priori depends on R, is in fact independent of R.

KIRK E. KLASSEN 24 (Hi) $((),*) = Id for allt. (iv)$(A, t) is a compact perturbation of the identity. Proof. 1. 1 of [KK4] shows that the projections to L(t) 0 PQ~0O vary analytically, and are pseudodifferential operators of order zero with the same symbol. Since F(to) = Id, F(t) is a compact perturbation of the identity. 1 as described in the paragraphs preceding the Proposition shows F(t) is analytic. 2. 1, and are left to the reader. For the last property, consider first the Sobolev L\ norm on sections of E defined by Hl>iMli=5>*l 2 (i + ^)k k 2 Then the inclusion L\{E) C L (E) is compact by Rellich's theorem.