# Analytic Deformations of the Spectrum of a Family of Dirac - download pdf or read online

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, rather, how this spectrum varies less than an analytic perturbation of the operator. forms 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 unlimited collar hooked up to its boundary.

The unifying proposal in the back of the research of those kinds of spectra is the concept of sure "eigenvalue-Lagrangians" within the symplectic area $L^2(\partial M)$, an concept because 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 different types of spectra which go through 0 behave in primarily an analogous manner (to first non-vanishing order). at times, this results in topological algorithms for computing spectral movement.

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By definition, (j)(t) respects (L(t), R) if and only if a(t) £ L(t) for all t. 3. Recall that K(0,0) C W(0) denotes the limiting values of extended L2 solutions to D(0)x = 0. Since A(0) = 0, it follows from the definitions that a(0) £ K(0,0)C\L(0). In particular, if L(t) is chosen so that L(0) is transverse to JFf(0,0), then a(0) = 0, so that the mth derivative of 4> also exponentially approaches the mth derivative of a, even if A^m^ ^ 0. Proof. ) Then n a(t) = ]£(afc(t)tyfc(t) - i^-ib(t)) + (bk{t)(iljk(t) + ty-fcW).

The preceding discussion also shows that the phenomena of the L2(X(oo)) point spectrum being absorbed into the continuous spectrum under deformations is can be viewed as a consequence of the discontinuity of the symplectic reduction map Lag(L2(E)) —> Lag(H). Indeed, a type 1 extended L2 eigenvector is type 1 with respect to any choice of L(t). Therefore, if L(0) is transverse to if (0,0), the extended L2 kernel of D(0) can be decomposed orthogonally into type 1 and type 2 eigenvectors, and the type 2 eigenvectors are type 2 for any choice of L(t) provided L(0) is transverse to if(0,0).

G. 2). 1 Discreteness near zero of extended L2 eigenvalues We now prove that the extended L2 eigenvalues which respect (L, R) form a discrete set near 0. To avoid further complications and since we are interested in eigenvalues near zero we now make the following restriction. Assumption. e. |A| < / i n + i . 1 DEFINITION. Define the set of small extended L2 X-eigenvalues which respect (L, R) to be E* = {A e (-/. n+1 /2,/i n+1 /2) I iV* n (L e p+) ? o}. To each point A € E ^ one assigns its multiplicity which is the dimension of N*n{L®P+).