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

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The notation (-^)k will always denote the derivative at t = 0; also the notation f(m) m e a n s the mth derivative of / at t = 0. 4 LEMMA. Let 0(£), t e (—e, e) be a smooth path of extended L2 eigenvectors for D(t), so that D(t)(/)(t) = A(£)0(£), with A(0) = 0. Assume the dimension of the kernel of the tangential operator H{t) is independent oft. Fix R>0 and let a(t) =projn{t)((f)(t)lR). Suppose that the derivatives A ^ = 0 for k < m. Then there exists a constant 6 > 0 depending only on D(t), and a constant C > 0 so that II (^)mW)\u) - (jt)m(<*(t)) + * ( m ) (" - R)M0) \\LHY,E)< Ce~^-R\ The constant 6 can be taken to be half the smallest positive eigenvalue /Jn+i of the tangential operators A(t).

The kernel of Z(A, t) is isomorphic to to the intersection of p(M\(t)) with V, which in turn, is isomorphic to the intersection of N\(t) with L(t) 0P^~(£). Thus the {Xi(t)} parameterize S ^ ( t ) . (t)(t)n(V(BW) = M Ai ( t )(t)fl L(0) 0 P0+(0). Let Ti(t) = W ^ W ^ M * ) - Then nWeiVAi(t)(t)n(LW0P+(t)(t)) is an analytic path of vectors. Consequently the path 4>i(t) = QxiW n(t) e /Ci / 2 (A - M*)) is an analytic path of Xi(t) eigenvectors whose restriction to the boundary is n{t). The set {i(0)} is a basis of D(0) : C°°(E; L(0) 0 P 0 + (0)) -> C°°{E) • as desired.

Thus the first form is positive definite. To show the second is positive definite, we need to show that {v,-§*(A,0)t;} dX |(o,o) is non-negative for any v. 4. Since 3>(A,0)V>fc = Vk,\, one sees that A 0A 1(0,0) *(A0to. =//i(°)Wfc V ' ' + I 0 ^-fc iffe>n ' if 0 < Jfc0 Vktpk- Then aA '(°'0> £o 2/Xfe ^ = {XI "fcV'fc. 3. We assume for notational ease that R = 0 and drop the superscripts R. 4. We have: Nx(t) n (L(t) e p+(t)) * *(A,t)-\Nx(t)) n (L(o) e p+(o)).